diff --git a/lib/oga/css/lexer.rl b/lib/oga/css/lexer.rl
index aab61bc..b98d860 100644
--- a/lib/oga/css/lexer.rl
+++ b/lib/oga/css/lexer.rl
@@ -147,7 +147,7 @@ module Oga
         # Identifiers are used for element and attribute names. Identifiers have
         # to start with a letter.
 
-        identifier = [a-zA-Z*]+ [a-zA-Z\-_0-9]*;
+        identifier = '*' | [a-zA-Z]+ [a-zA-Z\-_0-9]*;
 
         action emit_identifier {
           emit(:T_IDENT, ts, te)
@@ -184,6 +184,23 @@ module Oga
           add_token(:T_INT, value)
         }
 
+        # Strings
+        #
+        # Strings can be single or double quoted. They are mainly used for
+        # attribute values.
+        #
+        dquote = '"';
+        squote = "'";
+
+        string_dquote = (dquote ^dquote* dquote);
+        string_squote = (squote ^squote* squote);
+
+        string = string_dquote | string_squote;
+
+        action emit_string {
+          emit(:T_STRING, ts + 1, te - 1)
+        }
+
         # Nth numbers
         #
         # These numbers are in the form of 2n+1 and are used for
@@ -236,6 +253,7 @@ module Oga
           nth_integer    => emit_nth_integer;
           nth_identifier => emit_nth_identifier;
           integer        => emit_integer;
+          string         => emit_string;
 
           any;
         *|;
diff --git a/lib/oga/css/parser.y b/lib/oga/css/parser.y
index 2db64bd..f724dfc 100644
--- a/lib/oga/css/parser.y
+++ b/lib/oga/css/parser.y
@@ -14,6 +14,7 @@ rule
   expression
     : path
     | node_test
+    | node_operators
     ;
 
   path_member
@@ -46,7 +47,39 @@ rule
     ;
 
   predicate
-    : T_LBRACK expression T_RBRACK { val[1] }
+    : T_LBRACK predicate_members T_RBRACK { val[1] }
+    ;
+
+  predicate_members
+    : expression { val[0] }
+    | operator   { val[0] }
+    ;
+
+  operator
+    : op_members T_EQ          op_members { s(:eq, val[0], val[2]) }
+    | op_members T_SPACE_IN    op_members { s(:space_in, val[0], val[2]) }
+    | op_members T_STARTS_WITH op_members { s(:starts_with, val[0], val[2]) }
+    | op_members T_ENDS_WITH   op_members { s(:ends_with, val[0], val[2]) }
+    | op_members T_IN          op_members { s(:in, val[0], val[2]) }
+    | op_members T_HYPHEN_IN   op_members { s(:hyphen_in, val[0],val[2]) }
+    ;
+
+  node_operators
+    : node_test T_CHILD     node_test { s(:child, val[0], val[2]) }
+    | node_test T_FOLLOWING node_test { s(:following, val[0], val[2]) }
+    | node_test T_FOLLOWING_DIRECT node_test
+      {
+        s(:following_direct, val[0], val[2])
+      }
+    ;
+
+  op_members
+    : node_test
+    | string
+    ;
+
+  string
+    : T_STRING { s(:string, val[0]) }
     ;
 end
 
diff --git a/spec/oga/css/lexer/operators_spec.rb b/spec/oga/css/lexer/operators_spec.rb
index 36130bf..b197181 100644
--- a/spec/oga/css/lexer/operators_spec.rb
+++ b/spec/oga/css/lexer/operators_spec.rb
@@ -22,6 +22,10 @@ describe Oga::CSS::Lexer do
       lex_css('*=').should == [[:T_IN, nil]]
     end
 
+    example 'lex an identifier followed by the *= operator' do
+      lex_css('foo *=').should == [[:T_IDENT, 'foo'], [:T_IN, nil]]
+    end
+
     example 'lex the |= operator' do
       lex_css('|=').should == [[:T_HYPHEN_IN, nil]]
     end
diff --git a/spec/oga/css/lexer/strings_spec.rb b/spec/oga/css/lexer/strings_spec.rb
new file mode 100644
index 0000000..7eb0f3f
--- /dev/null
+++ b/spec/oga/css/lexer/strings_spec.rb
@@ -0,0 +1,13 @@
+require 'spec_helper'
+
+describe Oga::CSS::Lexer do
+  context 'strings' do
+    example 'lex a single quoted string' do
+      lex_css("'foo'").should == [[:T_STRING, 'foo']]
+    end
+
+    example 'lex a double quoted string' do
+      lex_css('"foo"').should == [[:T_STRING, 'foo']]
+    end
+  end
+end
diff --git a/spec/oga/css/parser/operators_spec.rb b/spec/oga/css/parser/operators_spec.rb
new file mode 100644
index 0000000..e33e042
--- /dev/null
+++ b/spec/oga/css/parser/operators_spec.rb
@@ -0,0 +1,83 @@
+require 'spec_helper'
+
+describe Oga::CSS::Parser do
+  context 'operators' do
+    example 'parse the = operator' do
+      parse_css('x[a="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:eq, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the ~= operator' do
+      parse_css('x[a~="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:space_in, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the ^= operator' do
+      parse_css('x[a^="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:starts_with, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the $= operator' do
+      parse_css('x[a$="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:ends_with, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the *= operator' do
+      parse_css('x[a*="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:in, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the |= operator' do
+      parse_css('x[a|="b"]').should == s(
+        :test,
+        nil,
+        'x',
+        s(:hyphen_in, s(:test, nil, 'a'), s(:string, 'b'))
+      )
+    end
+
+    example 'parse the > operator' do
+      parse_css('x > y').should == s(
+        :child,
+        s(:test, nil, 'x'),
+        s(:test, nil, 'y')
+      )
+    end
+
+    example 'parse the + operator' do
+      parse_css('x + y').should == s(
+        :following_direct,
+        s(:test, nil, 'x'),
+        s(:test, nil, 'y')
+      )
+    end
+
+    example 'parse the ~ operator' do
+      parse_css('x ~ y').should == s(
+        :following,
+        s(:test, nil, 'x'),
+        s(:test, nil, 'y')
+      )
+    end
+  end
+end