Various processes in physics, biology or chemistry can be described by nonlinear partial differential equations and systems thereof. In this thesis, we deal with different types of models for crowded transport in the life and social sciences. In the beginning, we formally derive a nonlinear convection-diffusion model describing the evolution of intersecting pedestrian flows. Depending on the geometry of the domain and the direction of motion of the respective groups, we observe the formation of complex patterns like directional lanes or diagonal stripes. The corresponding system of partial differential equations can be interpreted as a (perturbed) gradient flow and existence as well as stability results are discussed. Next, we discuss the analysis of a cross-diffusion system of partial differential equations for a mixture of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally we present a numerical scheme for nonlinear continuity equations, which is based on a variational formulation as a gradient flow of an energy functional with respect to the Wasserstein distance. This scheme can be applied to a large class of nonlinear continuity equations and inherent features include positivity, energy decrease and mesh adaptation in the case of blow up densities or compactly supported solutions. We present various numerical illustrations showing the flexibility of the scheme.