How do contemporary Russian jet engines compare to Western ones?
200 kN range

They should not be necessarily behind. To compare apples to apples, the nearest similar thrust to the Aviadvigatel PS-90A1 (171 kN) is the General Electric CF6-6 (185 kN), but the CF6-6 is 20 years older, so it's not fair.

Comparing those two, the PA-90 has better specific fuel consumption in cruise, 0.595 vs. 0.646 lb/hr/lbf. Also the bypass ratios (BPR) are 4.4 and 5.76 respectively. So even with the lower BPR, which generally would imply a higher SFC, that's not the case.

                       PS-90A1     CF6-6       PW2040

Year                   1992        1971        1981
Thrust                 171 kN      185 kN      178 kN
SFC (cruise)           0.595       0.646       0.582
BPR                    4.4         5.76        5.9
Power-to-weight        5.9:1       5.08:1      5.57:1
OPR                    35.50       25.2        27.6


The low BPR but also low SFC, high overall pressure ratio (OPR), and high power-to-weight ratio, all hint at an advanced engine.

The A2 variant for instance was developed "in co-operation with Pratt &amp; Whitney." If I had to guess, it's because the PS-90 fills a gap in the thrust range, gone since ETOPS put the tris and quads in decline. Nowadays the most common thrusts are 100-120 kN for the narrow-bodies, and the latest big jets are 350+ kN.

Big ones

We haven't seen a twin-jet wide-body from Russia, say like the 777 or A330. If that day comes, and they decide to build an engine and not out-source it, then we can have a better comparison with what's available from the big three—GE, P&amp;W, and RR—in the high thrust range.

100 kN range

The smaller Aviadvigatel PD-14 is still in development, and it's in the 120 kN range. Wikipedia says its SFC is 0.526, which is comparable to the CFM56 (not the new LEAP model), I'm not sure how reliable that figure is, because the references linked don't show that figure, instead they say:


  10-15% [improvement] relative to other contemporary engines of the similar thrust range and application.


And a geared-turbofan offshoot is planned, so they want to compete with the PW1000G.

Military (trivia)

To this day the Kuznetsov NK-32 remains the "largest and most powerful engine ever fitted on a combat aircraft."



RE comment for TBO:


PS-90A:



  In 1997 the Supplement to the engine Type Certificate was received to approve the aircraft engine field operation based on technical condition without fixed mean time between overhauls (MTBO) rates.
  
  The PS-90A became the first Russian aircraft engine that accumulated over 9,000 hours without any removal - the engine installed on Aeroflot's Il-96-300 accumulated 9,936 hours (source).



CF6-80 (what I could find):



  ... many CF6-80s make it to two or more shop visits, with overhaul intervals ranging from 10,000 hr. for those in short-haul operations to 40,000 hr. for those in long-haul (source).


Again, comparable.

RE comment for Wikipedia SFC figures:

That's an issue with Wikipedia, they mix and match sea level and cruise SFC. The links/figures I already included in the answer are both for cruise SFC, which you can compare. There's a question here about how the SFC changes with altitude and speed. So, to compare, you need to compare figures from the same situation.



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