Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 240; Flight Crew Total 942.88; Flight Crew Type 942.88
Aircraft Equipment Problem Less Severe
Air Traffic Control Provided Assistance; Flight Crew Inflight Shutdown; Flight Crew Landed in Emergency Condition; Flight Crew Returned To Departure Airport; Flight Crew Requested ATC Assistance / Clarification
Aircraft
Right engine failed during climb out passing approximately FL330; about 10 to 15 mins after takeoff.  Failure appeared to be due to loss of oil pressure (0 oil pressure indication; RT ENG LP PUMP status message).  We immediately [advised ATC]. The flight was FO's leg and so we decided to have him continue flying while I handle checklists and coordination. We decided to turn around and start heading in the direction of ZZZ; the field we just departed from; one of the closest airports in point of time.  We secured the right engine; per the engine failure checklist.  We decided not to try restarting the engine because of no oil pressure.   We finished up checklists and coordinated with ATC and dispatch for a return to ZZZ for an otherwise uneventful single engine ILS landing in ZZZ.
Climbing through FL325; the autopilot kicked off and the airplane began yawing right. We saw that the #2 engine had failed; oil pressure was zero. I righted the aircraft and began a descent to FL200 while the Captain [advised ATC] and began the Engine Failure Checklist. We determined a relight was not warranted and secured the right engine. We accepted vectors from ZZZ Center and I began the process of diverting to ZZZ. We accomplished an uneventful single engine approach and landing to [Runway] XXR. The brakes were heating a little by the time we reached the gate; and we were only about 5 minutes after touchdown; so I requested that ramp chock the wheels and otherwise remain clear. Brake temps settled at 4.
B757 flight crew reported to the departure airport after the #2 engine failed during climb out.