This thesis deals with hypocoercive Fokker-Planck equations, in particular the existence of steady states and long-term behaviour. For fully parabolic Fokker-Planck equations, entropy methods have been established as reliable tools in the analysis of long-term behaviour, providing sharp decay rate estimates in cases where a unique steady state exists. We show how this method can be extended to deal with degenerate parabolic equations in the context of hypocoercivity. For linear drift coefficients, we give a full characterisation of the equations which possess unique steady states, provide a sharp decay rate for general admissable entropies, and give the full spectrum of the operator. Finally, we discuss how our results extend to the case of nonlinear drift coefficients. In a second part we discuss a possible entropy method for discrete open quantum systems in Lindblad form. We establish a connection to Fokker-Planck equations and discuss the essential problems encountered in this approach.