diff --git a/src/normalizer.ts b/src/normalizer.ts
index 88c2d492..f03f8847 100644
--- a/src/normalizer.ts
+++ b/src/normalizer.ts
@@ -131,6 +131,20 @@ rules.set('Make extends always an array, if it is defined', schema => {
   }
 })
 
+rules.set('Transform $defs to definitions', schema => {
+  if (schema.$defs) {
+    schema.definitions = schema.$defs
+    delete schema.$defs
+  }
+})
+
+rules.set('Transform const to singleton enum', schema => {
+  if (schema.const) {
+    schema.enum = [schema.const]
+    delete schema.const
+  }
+})
+
 export function normalize(rootSchema: LinkedJSONSchema, filename: string, options: Options): NormalizedJSONSchema {
   rules.forEach(rule => traverse(rootSchema, schema => rule(schema, filename, options)))
   return rootSchema as NormalizedJSONSchema
diff --git a/test/__snapshots__/test/test.ts.md b/test/__snapshots__/test/test.ts.md
index 56e8e749..cec0a479 100644
--- a/test/__snapshots__/test/test.ts.md
+++ b/test/__snapshots__/test/test.ts.md
@@ -7759,6 +7759,17 @@ Generated by [AVA](https://avajs.dev).
       "required": []␊
     }`
 
+## Normalize const to singleton enum
+
+> Snapshot 1
+
+    `{␊
+      "id": "foo",␊
+      "enum": [␊
+        "foobar"␊
+      ]␊
+    }`
+
 ## Default additionalProperties to true
 
 > Snapshot 1
@@ -7807,6 +7818,17 @@ Generated by [AVA](https://avajs.dev).
       "id": "DefaultID"␊
     }`
 
+## Normalize $defs to definitions
+
+> Snapshot 1
+
+    `{␊
+      "id": "foo",␊
+      "definitions": {␊
+        "bar": "baz"␊
+      }␊
+    }`
+
 ## Destructure unary types
 
 > Snapshot 1
diff --git a/test/__snapshots__/test/test.ts.snap b/test/__snapshots__/test/test.ts.snap
index 81a1c6eb..75fb7e08 100644
Binary files a/test/__snapshots__/test/test.ts.snap and b/test/__snapshots__/test/test.ts.snap differ
diff --git a/test/normalizer/constToEnum.json b/test/normalizer/constToEnum.json
new file mode 100644
index 00000000..9e3567c6
--- /dev/null
+++ b/test/normalizer/constToEnum.json
@@ -0,0 +1,11 @@
+{
+  "name": "Normalize const to singleton enum",
+  "in": {
+    "id": "foo",
+    "const": "foobar"
+  },
+  "out": {
+    "id": "foo",
+    "enum": ["foobar"]
+  }
+}
diff --git a/test/normalizer/defsToDefinitions.json b/test/normalizer/defsToDefinitions.json
new file mode 100644
index 00000000..050ca47f
--- /dev/null
+++ b/test/normalizer/defsToDefinitions.json
@@ -0,0 +1,15 @@
+{
+  "name": "Normalize $defs to definitions",
+  "in": {
+    "id": "foo",
+    "$defs" : {
+      "bar": "baz"
+    }
+  },
+  "out": {
+    "id": "foo",
+    "definitions" : {
+      "bar": "baz"
+    }
+  }
+}