In this thesis, we consider the mathematical formulation of thin liquid films, which are approximated by two-dimensional evolving surfaces. The resulting equation, which describes the fluid flow on such fluid films, is a vector-valued surface partial differential equation, namely the incompressible surface Navier-Stokes equation. We are interested in numerical approaches in order to solve this equation as well as the interaction with the underlying geometry. Thereby, the highly nonlinear coupling of the interfacial hydrodynamics and the geometric properties is analyzed in various examples. First, we derive the vorticity-stream function formulation, which circumvents the vector-valued structure of the equations. This allows us to use the standard surface finite element method to solve this alternative approach numerically This has already been considered in the literature, but for stationary surfaces. Here, we extend these basic ideas and propose the respective formulation on evolving surfaces. However, it turned out, that this approach is only valid for surfaces which are topologically invariant to a sphere. Thus, we consider the incompressible surface Navier-Stokes equation in its original form and propose a further approach, which is based on the reformulation in the Euclidean basis, the Chorin projection method and spatial discretization with the standard surface finite element method. This allows us to study the geometric effects on more general surfaces. Additionally, we present two extensions of the proposed approaches. The first extension takes the interaction of the interfacial hydrodynamics and the surrounding fluids into account, for which a model based on phase fields is derived. Finally, an approach for polar liquid crystals on evolving surfaces is proposed, which couples the incompressible surface Navier-Stokes equation to another vector-valued surface partial differential equation for their orientational ordering. All examples show the highly nonlinear coupling between topology, geometric properties, defect interactions, shape changes and hydrodynamics.