Abstract: In this paper, we discuss the model order reduction problem for descriptor systems, that is, systems with dynamics described by differential-algebraic equations. We focus on linear descriptor systems as a broad variety of methods for these exist, while model order reduction for nonlinear descriptor systems has not received sufficient attention up to now. Model order reduction for linear state-space systems has been a topic of research for about 50 years at this writing, and by now can be considered as a mature field. The extension to linear descriptor systems usually requires extra treatment of the constraints imposed by the algebraic part of the system. For almost all methods, this causes some technical difficulties, and these have only been thoroughly addressed in the last decade. We will focus on these developments in particular for the popular methods related to balanced truncation and rational interpolation. We will review efforts in extending these approaches to descriptor systems, and also add the extension of the so-called stochastic balanced truncation method to descriptor systems which so far cannot be found in the literature.