The main theorem of the paper is that compact metric spaces which are locally n-connected and which have cohomological dimension less than or equal to n for some n are precisely the spaces which are cell-like images of finite polyhedra. This is used to characterize metric spaces which are limits of finite polyhedra in certain topological moduli spaces introduced by Gromov. The upshot is that up to simple-homotopy, nearby polyhedra can be reconstructed from the limit points, so little information is lost by passing from the space to its closure. The final section discusses the problem of determining which metric spaces are limits of closed manifolds in Gromov's spaces.