This book presents a range of entropy methods for diffusive partial differential equations (PDEs) devised by many researchers over the past few decades. These methods allow us to understand the qualitative behavior of solutions to diffusive equations and Markov diffusion processes. Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers with an introduction to selected entropy methods found in the research literature. To highlight the core concepts, the results are not stated in the widest generality, and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is assumed). The text is also suitable for advanced master's and PhD students.