diff --git a/src/frontends/lean/builtin_cmds.cpp b/src/frontends/lean/builtin_cmds.cpp
index 3647d5a9b..22a006670 100644
--- a/src/frontends/lean/builtin_cmds.cpp
+++ b/src/frontends/lean/builtin_cmds.cpp
@@ -9,6 +9,18 @@ Author: Leonardo de Moura
 
 namespace lean {
 static name g_raw("raw");
+
+environment universe_cmd(parser & p) {
+    if (p.curr_is_identifier()) {
+        // TODO(Leo): take namespace and scopes into account
+        name n = p.get_name_val();
+        p.next();
+        return p.env().add_global_level(n);
+    } else {
+        throw parser_error("invalid universe declaration, identifier expected", p.cmd_pos());
+    }
+}
+
 environment print_cmd(parser & p) {
     if (p.curr() == scanner::token_kind::String) {
         p.regular_stream() << p.get_str_val() << "\n";
@@ -25,7 +37,8 @@ environment print_cmd(parser & p) {
 
 cmd_table init_cmd_table() {
     cmd_table r;
-    add_cmd(r, cmd_info("print", "print a string", print_cmd));
+    add_cmd(r, cmd_info("print",    "print a string", print_cmd));
+    add_cmd(r, cmd_info("universe", "declare a global universe level", universe_cmd));
     return r;
 }
 
diff --git a/src/frontends/lean/builtin_exprs.cpp b/src/frontends/lean/builtin_exprs.cpp
index 944b73673..8352aa984 100644
--- a/src/frontends/lean/builtin_exprs.cpp
+++ b/src/frontends/lean/builtin_exprs.cpp
@@ -5,9 +5,23 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 */
 #include "frontends/lean/builtin_exprs.h"
+#include "frontends/lean/parser.h"
 
 namespace lean {
 namespace notation {
+static name g_llevel_curly(".{");
+static name g_rcurly("}");
+
+static expr parse_Type(parser & p, unsigned, expr const *) {
+    if (p.curr_is_token(g_llevel_curly)) {
+        p.next();
+        level l = p.parse_level();
+        p.check_token_next(g_rcurly, "invalid Type expression, '}' expected");
+        return mk_sort(l);
+    } else {
+        return mk_sort(p.mk_new_level_param());
+    }
+}
 
 parse_table init_nud_table() {
     action Expr(mk_expr_action());
@@ -19,6 +33,7 @@ parse_table init_nud_table() {
     r.add({transition("(", Expr), transition(")", Skip)}, x0);
     r.add({transition("fun", Binders), transition(",", mk_scoped_expr_action(x0))}, x0);
     r.add({transition("Pi", Binders), transition(",", mk_scoped_expr_action(x0, 0, false))}, x0);
+    r.add({transition("Type", mk_ext_action(parse_Type))}, x0);
     return r;
 }
 
diff --git a/src/frontends/lean/parse_table.cpp b/src/frontends/lean/parse_table.cpp
index 5877b9699..e7af2fb60 100644
--- a/src/frontends/lean/parse_table.cpp
+++ b/src/frontends/lean/parse_table.cpp
@@ -145,7 +145,7 @@ action mk_exprs_action(name const & sep, expr const & rec, expr const & ini, boo
 action mk_scoped_expr_action(expr const & rec, unsigned rb, bool lambda) {
     return action(new scoped_expr_action_cell(rec, rb, lambda));
 }
-action mk_ext_parse_action(parse_fn const & fn) { return action(new ext_action_cell(fn)); }
+action mk_ext_action(parse_fn const & fn) { return action(new ext_action_cell(fn)); }
 
 struct parse_table::cell {
     bool                                                         m_nud;
diff --git a/src/frontends/lean/parse_table.h b/src/frontends/lean/parse_table.h
index 413b3bd71..54a422840 100644
--- a/src/frontends/lean/parse_table.h
+++ b/src/frontends/lean/parse_table.h
@@ -61,7 +61,7 @@ public:
     friend action mk_binder_action();
     friend action mk_binders_action();
     friend action mk_scoped_expr_action(expr const & rec, unsigned rbp, bool lambda);
-    friend action mk_ext_parse_action(parse_fn const & fn);
+    friend action mk_ext_action(parse_fn const & fn);
 
     action_kind kind() const;
     unsigned rbp() const;
@@ -81,7 +81,7 @@ action mk_exprs_action(name const & sep, expr const & rec, expr const & ini, boo
 action mk_binder_action();
 action mk_binders_action();
 action mk_scoped_expr_action(expr const & rec, unsigned rbp = 0, bool lambda = true);
-action mk_proc_action(parse_fn const & fn);
+action mk_ext_action(parse_fn const & fn);
 
 class transition {
     name           m_token;
diff --git a/src/frontends/lean/parser.cpp b/src/frontends/lean/parser.cpp
index adf0e485b..0172b18c5 100644
--- a/src/frontends/lean/parser.cpp
+++ b/src/frontends/lean/parser.cpp
@@ -174,6 +174,15 @@ bool parser::curr_is_token(name const & tk) const {
         get_token_info().value() == tk;
 }
 
+bool parser::curr_is_token_or_id(name const & tk) const {
+    if (curr() == scanner::token_kind::Keyword || curr() == scanner::token_kind::CommandKeyword)
+        return get_token_info().value() == tk;
+    else if (curr() == scanner::token_kind::Identifier)
+        return get_name_val() == tk;
+    else
+        return false;
+}
+
 void parser::check_token_next(name const & tk, char const * msg) {
     if (!curr_is_token(tk))
         throw parser_error(msg, pos());
@@ -209,6 +218,124 @@ static name g_lcurly("{");
 static name g_rcurly("}");
 static name g_lbracket("[");
 static name g_rbracket("]");
+static name g_add("+");
+static name g_max("max");
+static name g_imax("imax");
+static name g_cup("\u2294");
+static unsigned g_level_add_prec = 10;
+static unsigned g_level_cup_prec = 5;
+
+unsigned parser::parse_small_nat() {
+    auto p  = pos();
+    mpz val = get_num_val().get_numerator();
+    next();
+    lean_assert(val >= 0);
+    if (!val.is_unsigned_int())
+        throw parser_error("invalid level expression, value does not fit in a machine integer", p);
+    return val.get_unsigned_int();
+}
+
+static level lift(level l, unsigned k) {
+    while (k > 0) {
+        k--;
+        l = mk_succ(l);
+    }
+    return l;
+}
+
+unsigned parser::curr_level_lbp() const {
+    if (curr_is_token(g_cup))
+        return g_level_cup_prec;
+    else if (curr_is_token(g_add))
+        return g_level_add_prec;
+    else
+        return 0;
+}
+
+level parser::parse_max_imax(bool is_max) {
+    auto p = pos();
+    next();
+    buffer<level> lvls;
+    while (curr_is_identifier() || curr_is_numeral() || curr_is_token(g_lparen)) {
+        lvls.push_back(parse_level());
+    }
+    if (lvls.size() < 2)
+        throw parser_error("invalid level expression, max must have at least two arguments", p);
+    unsigned i = lvls.size() - 1;
+    level r = lvls[i];
+    while (i > 0) {
+        --i;
+        if (is_max)
+            r = mk_max(lvls[i], r);
+        else
+            r = mk_imax(lvls[i], r);
+    }
+    return r;
+}
+
+level parser::parse_level_id() {
+    auto p  = pos();
+    name id = get_name_val();
+    next();
+    auto it = m_local_level_decls.find(id);
+    if (it != m_local_level_decls.end())
+        return it->second.first;
+    if (m_env.is_global_level(id))
+        return mk_global_univ(id);
+    throw parser_error(sstream() << "unknown level '" << id << "'", p);
+}
+
+level parser::parse_level_nud() {
+    if (curr_is_token_or_id(g_max)) {
+        return parse_max_imax(true);
+    } else if (curr_is_token_or_id(g_imax)) {
+        return parse_max_imax(false);
+    } else if (curr_is_token(g_lparen)) {
+        next();
+        level l = parse_level();
+        check_token_next(g_rparen, "invalid level expression, ')' expected");
+        return l;
+    } else if (curr_is_numeral()) {
+        unsigned k = parse_small_nat();
+        return lift(level(), k);
+    } else if (curr_is_identifier()) {
+        return parse_level_id();
+    } else {
+        throw parser_error("invalid level expression", pos());
+    }
+}
+
+level parser::parse_level_led(level left) {
+    auto p = pos();
+    if (curr_is_token(g_cup)) {
+        next();
+        level right = parse_level(g_level_cup_prec);
+        return mk_max(left, right);
+    } else if (curr_is_token(g_add)) {
+        next();
+        if (curr_is_numeral()) {
+            unsigned k = parse_small_nat();
+            return lift(left, k);
+        } else {
+            throw parser_error("invalid level expression, right hand side of '+' (aka universe lift operator) must be a numeral", p);
+        }
+    } else {
+        throw parser_error("invalid level expression", p);
+    }
+}
+
+level parser::parse_level(unsigned rbp) {
+    level left = parse_level_nud();
+    while (rbp < curr_level_lbp()) {
+        left = parse_level_led(left);
+    }
+    return left;
+}
+
+level parser::mk_new_level_param() {
+    // TODO(Leo)
+    return level();
+}
 
 /** \brief Parse <tt>ID ':' expr</tt>, where the expression represents the type of the identifier. */
 parameter parser::parse_binder_core(binder_info const & bi) {
diff --git a/src/frontends/lean/parser.h b/src/frontends/lean/parser.h
index 8f5e99e1f..cf9844076 100644
--- a/src/frontends/lean/parser.h
+++ b/src/frontends/lean/parser.h
@@ -44,7 +44,9 @@ struct interrupt_parser {};
 
 class parser {
     typedef std::pair<expr, unsigned> local_entry;
+    typedef std::pair<level, unsigned> local_level_entry;
     typedef scoped_map<name, local_entry, name_hash, name_eq> local_decls;
+    typedef scoped_map<name, local_level_entry, name_hash, name_eq> local_level_decls;
 
     environment             m_env;
     io_state                m_ios;
@@ -56,6 +58,7 @@ class parser {
     scanner                 m_scanner;
     scanner::token_kind     m_curr;
     local_decls             m_local_decls;
+    local_level_decls       m_local_level_decls;
     pos_info                m_last_cmd_pos;
     pos_info                m_last_script_pos;
     unsigned                m_next_tag_idx;
@@ -94,10 +97,18 @@ class parser {
     cmd_table const & cmds() const { return cfg().m_cmds; }
     parse_table const & nud() const { return cfg().m_nud; }
     parse_table const & led() const { return cfg().m_led; }
+    /** \brief Return true if the current token is a keyword named \c tk or an identifier named \c tk */
+    bool curr_is_token_or_id(name const & tk) const;
 
     void add_local_entry(name const & n, expr const & e);
     void add_local_decl(expr const & t);
 
+    unsigned curr_level_lbp() const;
+    level parse_max_imax(bool is_max);
+    level parse_level_id();
+    level parse_level_nud();
+    level parse_level_led(level left);
+
     void parse_command();
     void parse_script(bool as_expr = false);
     bool parse_commands();
@@ -126,6 +137,10 @@ public:
     script_state * ss() const { return m_ss; }
 
     void parse_names(buffer<std::pair<pos_info, name>> & result);
+    unsigned parse_small_nat();
+
+    level parse_level(unsigned rbp = 0);
+    level mk_new_level_param();
 
     parameter parse_binder();
     void parse_binders(buffer<parameter> & r);
@@ -145,6 +160,7 @@ public:
     expr rec_save_pos(expr const & e, pos_info p);
     pos_info pos_of(expr const & e, pos_info default_pos);
     pos_info pos_of(expr const & e) { return pos_of(e, pos()); }
+    pos_info cmd_pos() const { return m_last_cmd_pos; }
 
     /** \brief Read the next token. */
     void scan() { m_curr = m_scanner.scan(m_env); }
@@ -152,9 +168,11 @@ public:
     scanner::token_kind curr() const { return m_curr; }
     /** \brief Return true iff the current token is an identifier */
     bool curr_is_identifier() const { return curr() == scanner::token_kind::Identifier; }
+    /** \brief Return true iff the current token is a numeral */
+    bool curr_is_numeral() const { return curr() == scanner::token_kind::Numeral; }
     /** \brief Read the next token if the current one is not End-of-file. */
     void next() { if (m_curr != scanner::token_kind::Eof) scan(); }
-    /** \brief Return true iff the current token is equal to \c tk */
+    /** \brief Return true iff the current token is a keyword (or command keyword) named \c tk */
     bool curr_is_token(name const & tk) const;
     /** \brief Check current token, and move to next characther, throw exception if current token is not \c tk. */
     void check_token_next(name const & tk, char const * msg);
diff --git a/src/frontends/lean/token_table.cpp b/src/frontends/lean/token_table.cpp
index 3597fdb46..ad9d4aeee 100644
--- a/src/frontends/lean/token_table.cpp
+++ b/src/frontends/lean/token_table.cpp
@@ -39,16 +39,19 @@ static char const * g_lambda_unicode = "\u03BB";
 static char const * g_pi_unicode     = "\u03A0";
 static char const * g_forall_unicode = "\u2200";
 static char const * g_arrow_unicode  = "\u2192";
+static char const * g_cup            = "\u2294";
+
 
 token_table init_token_table() {
     token_table t;
     char const * builtin[] = {"fun", "Pi", "let", "in", "have", "show", "by", "from", "(", ")", "{", "}", "[", "]",
                               ".{", "Type", "...", ",", ".", ":", "calc", ":=", "--", "(*", "(--", "->",
-                              "proof", "qed", "private", "raw", "Bool", nullptr};
+                              "proof", "qed", "private", "raw", "Bool", "+", g_cup, nullptr};
 
     char const * commands[] = {"theorem", "axiom", "variable", "definition", "evaluate", "check",
                                "print", "variables", "end", "namespace", "section", "import",
-                               "abbreviation", "inductive", "record", "structure", "module", nullptr};
+                               "abbreviation", "inductive", "record", "structure", "module", "universe",
+                               nullptr};
 
     std::pair<char const *, char const *> aliases[] =
         {{g_lambda_unicode, "fun"}, {"forall", "Pi"}, {g_forall_unicode, "Pi"}, {g_pi_unicode, "Pi"},
diff --git a/src/tests/frontends/lean/scanner.cpp b/src/tests/frontends/lean/scanner.cpp
index 2b4a6027a..3afe06f24 100644
--- a/src/tests/frontends/lean/scanner.cpp
+++ b/src/tests/frontends/lean/scanner.cpp
@@ -147,12 +147,12 @@ static void tst2() {
     // check_name("x.10", name(name("x"), 10));
     check_name("x.bla", name({"x", "bla"}));
 
-    scan_error("+++");
+    scan_error("***");
     environment env;
     token_table s = mk_default_token_table();
-    s = add_token(s, "+++", "tplus");
+    s = add_token(s, "***", "tplus");
     env = update_token_table(env, s);
-    check_keyword("+++", "tplus", env);
+    check_keyword("***", "tplus", env);
     s = add_token(s, "+", "plus");
     env = update_token_table(env, s);
     check("x+y", {tk::Identifier, tk::Keyword, tk::Identifier}, env);
@@ -188,7 +188,7 @@ static void tst3() {
     scan_error("\"\\");
     scan_error("\"\\a");
     scan("\"\naaa\"");
-    scan_error("foo.+ 01");
+    scan_error("foo.* 01");
     scan("10.0.");
     scan("{ } . forall exists let in \u2200 := _");
 }
diff --git a/tests/lean/run/t2.lean b/tests/lean/run/t2.lean
new file mode 100644
index 000000000..a47f1120d
--- /dev/null
+++ b/tests/lean/run/t2.lean
@@ -0,0 +1,9 @@
+universe l
+universe u
+print raw Type
+print raw Type.{1}
+print raw Type.{2}
+print raw Type.{l+1}
+print raw Type.{max l u 1}
+print raw Type.{imax l+1 u 1}
+print raw Type.{imax l+1 l u}
\ No newline at end of file
diff --git a/tests/lua/token_table.lua b/tests/lua/token_table.lua
index 890dbb9a5..6b6b2d251 100644
--- a/tests/lua/token_table.lua
+++ b/tests/lua/token_table.lua
@@ -22,8 +22,8 @@ s = s:add_command_token("tast",  "tst2")
 s = s:add_command_token("tests", "tst3")
 s = s:add_command_token("fests", "tst4")
 s = s:add_command_token("tes",   "tst5")
-s = s:add_token("++", "++", 65)
-s = s:add_token("++-", "plusminus")
+s = s:add_token("**", "**", 65)
+s = s:add_token("**-", "ttimeminus")
 assert(token_table_size(s) == 7)
 display_token_table(s)