Petri nets are the most studied and well-regarded model for concurrent, parallel computations. This textbook presents, for the first time, numerous results from the original literature on impossibilities, possibilities, and the complexity of the expressive means of Petri nets in a didactic manner, providing detailed insights for a broader audience. All techniques and mathematical terms necessary for the proofs are explained. Thus, the book is aimed at both students and educators as well as researchers. The content focuses not only on presenting the fundamental concepts and their interrelations but also on an algorithm for the reachability problem, the expressiveness of various computational concepts, selected questions regarding decidability and complexity, as well as Petri net semantics through languages and partially ordered sets.