Abstract Both SAT and #SAT can represent difficult problems in seemingly dissimilar areas such as planning, verification, and probabilistic inference. Here, we examine an expressive new language, #?SAT, that generalizes both of these languages. #?SAT problems require counting the number of satisfiable formulas in a concisely-describable set of existentially-quantified, propositional formulas. We characterize the expressiveness and worst-case difficulty of #?SAT by proving it is complete for the complexity class #P NP [1] , and relating this class to more familiar complexity classes. We also experiment with three new general-purpose #?SAT solvers on a battery of problem distributions including a simple logistics domain. Our experiments show that, despite the formidable worst-case complexity of #P NP [1] , many of the instances can be solved efficiently by noticing and exploiting a particular type of frequent structure.