The most official answer is: The Ural Mountains. For the majority of people, this is the dividing line between European Russia and Asian Russia.

Georgia is an interesting case, because it is to the east of Istanbul, but is still relatively west of the Urals. It is basically on the border of the two continents. It in essence is... both. Even culturally it is both European in some ways and Asian in others, much more so than Russia.

I believe scientifically speaking, Georgia is located in Asia. However, even the Soviet Union debated if Georgia was Asian or European. Ultimately deciding that it was in Asia, but was European culturally.