As have most of the former Eastern bloc carriers (LOT, CSA, MALEV, Balkan, etc.), Aeroflot - Russian International Airlines has had to come to terms with the fact that Soviet types do not meet current standards of passenger comfort and efficiency. With the exception of the IL62, Aeroflot had no long-range passenger aircraft available until the introduction of the Airbus A310-300 into service in 1992. In addition to its 4 A310-300s (now 8, I believe), Aeroflot took delivery of the Ilyushin IL96-300, the first Soviet era fly-by-wire, high-bypass turbofan-powered widebody aircraft, in 1993. This plane has had many difficulties in the area of engine reliability, however, which led Aeroflot to lease two Boeing 767-300ERs in 1997. Aeroflot is supplementing these two aircraft with a further 4, already in service. In order to provide high capacity for routes to Asia, ARIA also leases two Boeing 777-200s, and in the interest of fleet modernisation and passenger comfort, introduced the Boeing 737-400 on routes to Western Europe, replacing many TU134 and TU154 aircraft on those flights.
With regards to the IL86, it was originally designed to be broadly equivalent to the Boeing 747, but did not have high-bypass turbofans. The lack of powerful, efficient engines caused the IL86 to be severely limited in range - only about 2,500 miles per leg without refueling. Therefore, there has been little or no interest in the IL86 from airlines other than Aeroflot. The IL86 does serve Aeroflot well on shorter-range high-capacity routes, such as the heavily traveled summer resort flights (St. Petersburg - Sochi, Moscow - Sochi, St. Petersburg - Simferopol', Moscow - Simferopol', etc.), so the aircraft still has a place in the airline's fleet.
I think that as production increases of the TU204 and IL96M/T, Aeroflot - Russian International Airlines will introduce them into service, replacing some, if not all Western types it currently operates. It is certain the IL96M will see service on both domestic and international routes, replacing the IL86 on routes that aircraft currently flies, as well as providing non-stop capabilities to Asia and North and South America.