This thesis is dedicated to the study of the qualitative properties of many-particle systems. We specifically consider those systems that can be described by a family of interacting diffusion processes, similar to the ones considered by Henry P. McKean. We study the behaviour of these systems in the limit as the number of particles $N$ tends to $\infty$. Their asymptotic behaviour in the mean field limit is described by nonlocal parabolic partial differential equations (PDEs) that are characterised by a competition between a linear or nonlinear diffusion term and a nonlocal interaction/aggregation term. These equations share the common feature of being gradient flows of a certain free energy functional, $E$, on the space of probability measures.

We provide a characterisation of both the set of stationary solutions of these nonlocal mean field PDEs and the set of minimisers of the associated energy, $E$. We demonstrate how these sets vary as the relative strength of the diffusion and aggregation terms is varied. This variation corresponds to the phenomenon of phase transitions. We also provide conditions on the interaction and confining potentials under which the minimisers of the energy $E$ have certain symmetries. Furthermore, we establish conditions under which the free energy $E$ possesses a saddle point and use this, along with an appropriate large deviations principle, to establish bounds on the escape probabilities of the underlying interacting particle systems. Finally, we study the effect of these phase transitions on the behaviour of the underlying particle system under certain distinguished scaling limits. Namely, we study the mean field limit $N \to \infty$ and the diffusive limit $\eps \to 0$ and show that, in general, these two limits do not commute in the presence of a phase transition.