This survey is the expanded version of my talk at the AMS meeting in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.

I would like to express my deep gratitude to my advisor Igor Frenkel for his guidance and encouragement throughout my study at Yale. Most of what I know about representation theory, mathematical physics and special functions comes from him. Also, I want to thank my friend and co-author Pavel Etingof; most of the results in this dissertation are based on our joint work, and would never be written without him; the idea of systematic study of the weighted traces of intertwiners belongs to him. His enthusiasm and new ideas often helped to find a new way when I was ready to give up. My understanding of the subjects discussed in this dissertation benefited a lot from fruitful discussions with other people, and among them Ivan Cherednik, Jintai