In the mathematical theory of viscous incompressible fluids the fundamental model is provided by the incompressible Navier-Stokes equations. For the last 20 years the analysis of the Navier-Stokes equations in the exterior of a moving and rotating obstacle has attracted particular attention. The main difficulty in this context arises from the fact that a linear coordinate transformation, which is used to handle the moving domain, results in transformed fluid equations that contain a drift term with linearly growing, hence unbounded coefficients. In this thesis the Navier-Stokes equations in the exterior of a moving, in particular rotating, obstacle are studied and the main emphasis is placed on the case where the obstacle undergoes a rotation described by a non-autonomous equation, and on fluids with variable density. In the non-autonomous case an appropriate functional analytic framework is adopted to show existence and uniqueness of local mild solutions. Moreover, based on the Faedo-Galerkin method and suitable a priori estimates, a well-posedness result for incompressible fluids with variable density is proved. For the proof new techniques needed to be developed. In particular, new elliptic estimates for the modified stationary Stokes problem with rotating effect are derived.