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abstract: 'We establish the global well-posedness of overdamped dynamical density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal flow and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With no-flux boundary conditions, we determine the well-posedness of the full nonlocal equations including two-body hydrodynamic interactions (HI) through the theory of Fredolm operators. Principally, this is done by rewriting the dynamics for the density $\varrho$ as a nonlocal Smoluchowski equation with a non-constant diffusion tensor $\bm{D}$ dependent on the diagonal part ($\bm{Z}_1$) of the HI tensor, and an effective drift $\bm{A}[\vec{a}]$ dependent on the off-diagonal part ($\bm{Z}_2$). We derive a scheme to uniquely construct the mean colloid flux $\vec{a}(\vec{r},t)$ in terms of eigenvectors of $\bm{D}$, show that the stationary density $\varrho(\vec{r})$ is independent of the HI tensors, as well as proving exponentially fast convergence to equilibrium. The stability of the equilibria $\varrho(\vec{r})$ is studied by considering the bounded (nonlocal) perturbation of the differential (local) part of the linearised operator. We show that the spectral properties of the full nonlocal operator with no-flux boundary conditions can differ considerably from those with periodic boundary conditions. We showcase our results by using the numerical 