This paper develops a class of probability models for configurations of interacting points in a domain. The distributions depend on a function which may be viewed as giving the potential energy of the configurations. Examples include models for interaction in a spatial region and on lattices and graphs. New models in the general class arise naturally, an example being a spatial model for points of different categories. Some general methods, including series expansions and a simulation method known from statistical mechanics, can be useful for many of these models. Several applications of this kind are considered, and some connections between the models and with the statistical mechanics are explored.