Classical density functional theory (DFT) is a sophisticated and versatile tool to tackle fundamental physical questions regarding the behavior of various types of equilibrium fluids through a variational principle with respect to the probability density to find a particle at a certain position. The framework of DFT has a deep mathematical foundation provided by a rigorous theorem relating the density of the fluid and the external potential acting on the particles: the knowledge of the density uniquely determines the external potential. The aim of this research project is to establish the mathematical background of a dynamical DFT (DDFT), which we understand in the most general ma- thematical sense as a differential equation for the time-dependent density. The central question is whether the different random external influences on the nonequilibrium behavior of a fluid can be uniquely characterized alone in terms of the density which solves the DDFT equation, in generalization of the uniqueness theorem in equilibrium. This purely mathematical task is independent of explicit approximations required for practical purposes. It will be complemented by developing and employing novel DDFT methods to describe the complex interplay of different intrinsic and extrinsic forces acting on the particles. A particular focus will lie on nonequilibrium fluid behavior in porous (random) media.