This survey aims to present a comprehensive and systematic synthesis of concepts and results on the minimal state space realization problem for positive, linear, time-invariant systems. Positive systems are systems for which the state and the output are always non-negative for any non-negative initial state and input. They are used to model phenomena in which the variables must take non-negative values due to the nature of the underlying physical system. Restricting the state–space realization to positive systems makes the problem extremely different and much more difficult than that for ordinary systems. Indeed, a minimal positive realization may have a dimension even much greater than the order of the transfer function it realizes. Although the problem of finding a finite-dimensional positive state–space realization of a given transfer function has been solved, the characterization of minimality for positive systems is still an open problem. This survey introduces the reader to different aspects of the problem and presents the mathematical approaches used to tackle it as well as some relevant related problems. Moreover, some partial results are presented. Finally, a comprehensive bibliography on positive systems, organized by topics, is provided.