diff --git a/spec-dtslint/operators/exhaust-spec.ts b/spec-dtslint/operators/exhaust-spec.ts
index 85c72c7390..e4ffc3d145 100644
--- a/spec-dtslint/operators/exhaust-spec.ts
+++ b/spec-dtslint/operators/exhaust-spec.ts
@@ -8,3 +8,14 @@ it('should infer correctly', () => {
 it('should enforce types', () => {
   const o = of(1, 2, 3).pipe(exhaust()); // $ExpectError
 });
+
+// TODO(benlesh): The following test fails for TypeScript 3.1, but passes in TypeScript 3.2
+// I'm unsure what we need to do to get this so it ignores the TS 3.1 failure, as that's a bug
+// in TypeScript, and this is properly typed now.
+
+// it('should support union types', () => {
+//   const a = Math.random() > 0.5 ? of(123) : of('abc');
+//   const b = Math.random() > 0.5 ? of(123) : of('abc');
+//   const source = of(a, b);
+//   const o = source.pipe(exhaust()); // $ExpectType Observable<string | number>
+// });
diff --git a/spec-dtslint/operators/exhaustMap-spec.ts b/spec-dtslint/operators/exhaustMap-spec.ts
index 47f9114cef..c82dae6606 100644
--- a/spec-dtslint/operators/exhaustMap-spec.ts
+++ b/spec-dtslint/operators/exhaustMap-spec.ts
@@ -29,6 +29,10 @@ it('should support an undefined resultSelector', () => {
   const o = of(1, 2, 3).pipe(exhaustMap(p => of(Boolean(p)), undefined)); // $ExpectType Observable<boolean>
 });
 
+it('should report projections to union types', () => {
+  const o = of(Math.random()).pipe(exhaustMap(n => n > 0.5 ? of('life') : of(42))); // $ExpectType Observable<string | number>
+});
+
 it('should enforce types', () => {
   const o = of(1, 2, 3).pipe(exhaustMap()); // $ExpectError
 });
diff --git a/src/internal/operators/exhaustMap.ts b/src/internal/operators/exhaustMap.ts
index acecff4029..f54f59bc8f 100644
--- a/src/internal/operators/exhaustMap.ts
+++ b/src/internal/operators/exhaustMap.ts
@@ -5,14 +5,14 @@ import { Subscription } from '../Subscription';
 import { OuterSubscriber } from '../OuterSubscriber';
 import { InnerSubscriber } from '../InnerSubscriber';
 import { subscribeToResult } from '../util/subscribeToResult';
-import { ObservableInput, OperatorFunction } from '../types';
+import { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';
 import { map } from './map';
 import { from } from '../observable/from';
 
 /* tslint:disable:max-line-length */
-export function exhaustMap<T, R>(project: (value: T, index: number) => ObservableInput<R>): OperatorFunction<T, R>;
+export function exhaustMap<T, O extends ObservableInput<any>>(project: (value: T, index: number) => O): OperatorFunction<T, ObservedValueOf<O>>;
 /** @deprecated resultSelector is no longer supported. Use inner map instead. */
-export function exhaustMap<T, R>(project: (value: T, index: number) => ObservableInput<R>, resultSelector: undefined): OperatorFunction<T, R>;
+export function exhaustMap<T, O extends ObservableInput<any>>(project: (value: T, index: number) => O, resultSelector: undefined): OperatorFunction<T, ObservedValueOf<O>>;
 /** @deprecated resultSelector is no longer supported. Use inner map instead. */
 export function exhaustMap<T, I, R>(project: (value: T, index: number) => ObservableInput<I>, resultSelector: (outerValue: T, innerValue: I, outerIndex: number, innerIndex: number) => R): OperatorFunction<T, R>;
 /* tslint:enable:max-line-length */
@@ -59,15 +59,15 @@ export function exhaustMap<T, I, R>(project: (value: T, index: number) => Observ
  * @method exhaustMap
  * @owner Observable
  */
-export function exhaustMap<T, I, R>(
-  project: (value: T, index: number) => ObservableInput<I>,
-  resultSelector?: (outerValue: T, innerValue: I, outerIndex: number, innerIndex: number) => R,
-): OperatorFunction<T, I|R> {
+export function exhaustMap<T, R, O extends ObservableInput<any>>(
+  project: (value: T, index: number) => O,
+  resultSelector?: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R,
+): OperatorFunction<T, ObservedValueOf<O>|R> {
   if (resultSelector) {
     // DEPRECATED PATH
     return (source: Observable<T>) => source.pipe(
       exhaustMap((a, i) => from(project(a, i)).pipe(
-        map((b, ii) => resultSelector(a, b, i, ii)),
+        map((b: any, ii: any) => resultSelector(a, b, i, ii)),
       )),
     );
   }