In this memoir we consider the Dirichlet problem for parabolic operators in a half space with singular drift terms. In chapter I we begin the study of a parabolic PDE modeled on the pullback of the heat equation in certain time varying domains considered by Lewis-Murray and Hofmann-Lewis. In chapter II we obtain mutual absolute continuity of parabolic measure and Lebesgue measure on the boundary of this halfspace and also that the $L^q(R^n)$ Dirichlet problem for these PDE's has a solution when $q$ is large enough. In chapter III we prove an analogue of a theorem of Fefferman, Kenig, and Pipher for certain parabolic PDE's with singular drift terms. Each of the chapters that comprise this memoir has its own numbering system and list of references.