I've seen 'сочувствие' used as a synonym for both sympathy and empathy. Does it correspond to either one in particular or to both in some way?
I'd include another word - compassion - to the list of synonyms. This page describes this term as a positive one, however in Russian the phrase "Я тебе сочуствую" can be used to ridicule somebody, but not in a harsh way.
empathy is an interesting word for a Russian. I learned it when I was around 18 - in English. It was a new concept for me. What I understood English speakers meant by empathy was neither сочувствие nor сопереживание to me. To them empathy was a desired quality that one should strive to develop, a gift, and in my mind I agreed with them. But when I thought of сочувствие, it was more along the lines of feeling sorry for a loser that does not know how to take care of his life. сопереживание was a fancy literary equivalent of сочувствие of sorts - it was one of those words that I understood more through root analysis (suffer together) than through the feel of usage. Granted, my vision of the Russian language was incomplete due to my age and some lack of emotional maturity in some aspects, but the reality is that when you speak to Russians you will find many people in a similar situation.
It is important to understand that going from culture to culture some concepts have one precise word that describes them in one, but will require sometimes a paragraph or more to properly communicate in the other. I think the English empathy going to Russian is one of them. Not to be outdone, we have some words that require more than one word to translate into English. I keep running into those while reading Russian literature with my children. Some colorful examples I had to explain recently that perhaps will help one understand why we do not have a good word for empathy:
расстрел - execution by firing squad
автоматчик - a soldier with an automatic rifle ready to use
Сочувствие is more about sympathy, but it also may mean an empathy, it depends on a context. The company I work at used to deal with Russian texts rather often, it is very difficult and the meanings of different words and phrases often depend on the emotional message of the sentence. We used to order translations from BNG or Technolex Translations, because it is simply impossible to translate the whole doc perfectly :)