We consider two variations on the optimal transportation problem where the particles/agents being transported interact with each other. For instance, imagine the problem of moving a collection of boxes from one configuration to another, where all the boxes move in unison and must avoid each other. As we shall show, these problems can be posed as quadratic optimization problems in the space of probability measures over the space of paths. Although the resulting optimization problem is not always convex, one can show existence and even uniqueness for some types of interactions. Moreover, we show these problems admit a fluid mechanics formulation in the style of Benamou and Brenier. This talk is based on works in collaboration with René Cabrera (UT Austin) and Jacob Homerosky (Texas State).