diff --git a/lib/dry/types/compiler.rb b/lib/dry/types/compiler.rb
index 39a254ef..9f8f74b2 100644
--- a/lib/dry/types/compiler.rb
+++ b/lib/dry/types/compiler.rb
@@ -113,6 +113,16 @@ def visit_any(meta)
         registry["any"].meta(meta)
       end
 
+      def visit_intersection(node)
+        *types, meta = node
+        types.map { |type| visit(type) }.reduce(:&).meta(meta)
+      end
+
+      def visit_implication(node)
+        *types, meta = node
+        types.map { |type| visit(type) }.reduce(:>).meta(meta)
+      end
+
       def compile_fn(fn)
         type, *node = fn
 
diff --git a/spec/dry/types/compiler_spec.rb b/spec/dry/types/compiler_spec.rb
index 48cb126d..58d28e94 100644
--- a/spec/dry/types/compiler_spec.rb
+++ b/spec/dry/types/compiler_spec.rb
@@ -377,4 +377,40 @@
       expect(type).to eql(source)
     end
   end
+
+  context "composites" do
+    let(:strict_nil_ast) do
+      [:constrained,
+       [[:nominal, [NilClass, {}]],
+        [:predicate, [:type?, [[:type, NilClass], [:input, Undefined]]]]]]
+    end
+
+    let(:strict_integer_ast) do
+      [:constrained,
+       [[:nominal, [Integer, {}]],
+        [:predicate, [:type?, [[:type, Integer], [:input, Undefined]]]]]]
+    end
+
+    let(:any_numeric_ast) do
+      [:constrained, [[:any, {}], [:predicate, [:type?, [[:type, Numeric], [:input, Undefined]]]]]]
+    end
+
+    it 'builds a sum' do
+      ast = [:sum, [strict_nil_ast, strict_integer_ast, {}]]
+      type = compiler.(ast)
+      expect(type).to eql(Dry::Types['integer'].optional)
+    end
+
+    it 'builds an implication' do
+      ast = [:implication, [any_numeric_ast, strict_integer_ast, {}]]
+      type = compiler.(ast)
+      expect(type).to eql(Dry::Types['any'].constrained(type: Numeric) > Dry::Types['integer'])
+    end
+
+    it 'builds an intersection' do
+      ast = [:intersection, [any_numeric_ast, strict_integer_ast, {}]]
+      type = compiler.(ast)
+      expect(type).to eql(Dry::Types['any'].constrained(type: Numeric) & Dry::Types['integer'])
+    end
+  end
 end
diff --git a/spec/dry/types/implication_spec.rb b/spec/dry/types/implication_spec.rb
index 5170949e..0a80b443 100644
--- a/spec/dry/types/implication_spec.rb
+++ b/spec/dry/types/implication_spec.rb
@@ -216,15 +216,14 @@
   end
 
   describe "#meta" do
-    context "optional types" do
-      let(:meta) { {foo: :bar} }
+    let(:meta) { {foo: :bar} }
 
-      subject(:type) { t::Nominal::String.optional }
+    subject(:type) { t::Nominal::Hash > t.Hash(foo: t::Nominal::Integer) }
 
-      it "uses meta from the right branch" do
-        expect(type.meta(meta).meta).to eql(meta)
-        expect(type.meta(meta).right.meta).to eql(meta)
-      end
+    it "has no special meta handling" do
+      expect(type.meta(meta).meta).to eql(meta)
+      expect(type.meta(meta).left.meta).to eql({})
+      expect(type.meta(meta).right.meta).to eql({})
     end
   end
 end
diff --git a/spec/dry/types/intersection_spec.rb b/spec/dry/types/intersection_spec.rb
index 10ceba2b..9c44faf2 100644
--- a/spec/dry/types/intersection_spec.rb
+++ b/spec/dry/types/intersection_spec.rb
@@ -231,15 +231,14 @@
   end
 
   describe "#meta" do
-    context "optional types" do
-      let(:meta) { {foo: :bar} }
+    let(:meta) { {foo: :bar} }
 
-      subject(:type) { Dry::Types["string"].optional }
+    subject(:type) { t::Nominal::Hash & t.Hash(foo: t::Nominal::Integer) }
 
-      it "uses meta from the right branch" do
-        expect(type.meta(meta).meta).to eql(meta)
-        expect(type.meta(meta).right.meta).to eql(meta)
-      end
+    it "has no special meta handling" do
+      expect(type.meta(meta).meta).to eql(meta)
+      expect(type.meta(meta).left.meta).to eql({})
+      expect(type.meta(meta).right.meta).to eql({})
     end
   end
 end