Twenty Open Problems in Enumeration of Matchings
  This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems in Enumeration of Matchings: Progress Report" (also available from this server as math.CO/9801061). NOTE: This article has now been superseded by math.CO/9904150. 
James Propp (MIT)