Can anyone suggest a good and recent survey on counting problems and/or problems that are #P. 
Survey on #P and/or counting problems
cc.complexity theory;reference request;counting complexity
L. Fortnow. Counting complexity. In L. Hemaspaandra and A. Selman, editors, Complexity Theory Retrospective II, pages 81-107. Springer, 1997This gives more of the structural complexity point of view (complexity classes, oracles, etc.), and discusses other classes related to #P. Although it's from almost 15 years ago, it's really not that out of date in terms of results.