Explicit criteria for the existence of geometric structures on manifolds are known in specific cases and may be understood as special solutions to a certain problem in classical obstruction theory. Adopting Cartan's (and Ehresmann's) views on differential geometry, geometric structures may be identified with multiplicative connections on transitive Lie groupoids: we are thus led to the natural question, what can you say in general about the existence (and deformation) of multiplicative connections on Lie groupoids? The purpose of this talk is to describe how, on condition that one confines attention to proper Lie groupoids, the general existence problem can be reduced to a simpler one which is analogous in many ways to the classical obstructiontheoretic problem mentioned at the beginning; the role of properness is related to linearizability, although the results that we are going to describe are not (and probably cannot be) deduced from the classical linearization theorems for proper Lie groupoids.
