This text gives an essentially self-contained exposition of the basic theory of marked point processes, developing randomly over time, and shows how this theory may be used to treat piecewise deterministic stochastic processes in continuous time.The focus is on point processes that generate only finitely many points in finite time intervals, resulting, in particular, in piecewise deterministic processes with "few jumps. The point processes are constructed from scratch and their distributions characterized using compensating measures and martingale structures. Piecewise deterministic processes are defined and identified with certain marked point processes, and this identification is used to construct and study a large class of piecewise deterministic Markov processes, whether time homogeneous or not.The second part of the book shows how the theory developed earlier is used to analyze various models in statistics and applied probability with examples from survival analysis, branching processes, risk theory, finance, queueing theory and sports (soccer). Graduate students, postdocs and researchers in the above-mentioned areas will find this text an excellent resource, requiring only a solid foundation in probability theory and measure theory as well as basic knowledge of stochastic processes and martingales.