Positive systems arise naturally in many areas of biology or medicine, where the variables represent nonnegative quantities such as concen-trations or population numbers of bacteria or cells. One encounters such systems also in economy, where the aim is to predict prices or productions. With this work we aim to lay the theoretical foundation for understanding positive descriptor systems in the continuous-time and in the discrete-time case. We present a cohesive framework that allows to generalise many results for standard positive systems to the descriptor case. We define and characterise positive descriptor sys-tems and analyse their properties. We establish two fundamental results, pertinent to the area of nonnegative matrix theory, that play a crucial role in the analysis of such systems. Furthermore, we treat cen-tral topics from systems and control theory, such as stability proper-ties, existence of doubly nonnegative solutions of generalised Lyapu-nov equations and positivity preserving model reduction. The present work is directed at students or professionals seeking an introduction into the field of positive descriptor systems or aiming to conduct further research in this area.