Back when airliners had flight engineers, what were their functions?

The progression of command shows it could happen that the Flight Engineer might end up at the controls. After the designated relief pilot (after the second officer).

As stated elsewhere the flight engineer for the most part monitored how the systems were functioning. Also they made sure that the engines were at the power setting desired by the pilot. Also functioning as a navigator and monitoring things such as air speed and altitude.

No. They are NEVER   Officially required to fly. First officer (:"co-pilot")  and Pilot are.

WRONG .Again ReRugged..

NO old man. Nav is First Officer's responsibility. These instruments are NOT  on FE's board. He is NOT required to Officially  fly--although he Might have too if Both Capn and FO pass out..

FE monitors engines, electrical. APU, etc. The runnin of the plane, not the   flyin  of it. I've sat in FE seat in a  737 in class;  monitored frequency.,  voltage, etc,.

REF: IN an A & P class. One  instructor  Was a FE., SSgt and A & P rated in USAF. .  He was NOT required to have a  Com. pilots license and multi-engine rating.

Monitor and control systems.  That was analog.  Now it is digital (computer and sensors).

Usually the 3rd officer (who actually goes through the same flight schooling as the pilot, etc. (S)he monitors flight systems that aren't auto regulated, and can fly the plane if called to do so. 

As I understand it, their duty was to monitor engine performance and other mechanical matters. 

Modern engines are fitted with electronic systems that control the various parameters to ensure that limits are not exceeded. For instance, as a function of the atmospheric conditions, chiefly air temperature, the maximum RPM of the engine has to be curbed so as to avoid making key component overheat.

Engine controllers (FADEC, for instance) do this automatically now, but 50 years ago, there was the need for someone to monitor the various temperatures and pressures and adjust accordingly. That was the flight engineer's function.

Wikipedia can answer your question in detail, lazybones. I know because I just checked and they have a page about flight engineers. There's no point in us explaining what you can easily learn on your own.