Moscow Mathematical Society Prize winner, 1970

Kargapolov Prize "for outstanding contributions to group theory", 1993

Honorary Silver Badge, Seminar "Algebra & Logic," Novosibirsk, 1987

J. Miller Professorship, University of Illinois at Urbana Champaign, March-April 1995

Russian Federation Presidential Award 1993-1996

INTAS Award 1994-1998

J. Soros Professorship, 1995-96 and 1998

Russian Fund (awarded to outstanding research groups), 1996-1997

Russian Fund (for fundamental research), 1996-1998

INTAS grant 99-1224, 2000-2003

Malcev Prize of the Russian Academy of Sciences for the series of papers in geometric and combinatorial group theory, 2000

Russian Fund for Fundamental Research, 1999-2001; grant 02-01-00170, 2002-2004; grant 05-01-00895, 2005-2007; grant 08-01-00573, 2008-2010.

NSF grant, DMS 0072307, 2000- 2003

NSF grant, DMS-0245600, 2003-2006

NSF Focus Research Grant, DMS-0455881, 2005-2008

NSF grant, DMS 0700811, 2007 - 2012, $948,596, PI (with Mark Sapir)