In an attempt to devise a general notion of model for spatial logic, I have been led to consider transition systems with an additional so-called spatial structure on the states, with both the transition and the spatial structures described in coalgebraic terms. The corresponding notion of bisimulation takes into account both structures and bisimilarity is equivalent to logical equivalence, thus extending Hennessy-Milner's result to the present framework. Transition systems with spatial structure can be seen as a noninterleaving model of concurrency, as there exist translations to and from a certain category of Petri nets, which constitute a pair of adjoint functors. The fragment of spatial logic considered will be related to coalgebraic modal logics, and it will be shown how the spatial operators are derived operators in a more primitive coalgebraic modal logic.