This text offers a mathematically rigorous 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 point processes are constructed from scratch with detailed proofs. The second part of the book addresses applications of the just developed theory; this analysis of various models in applied statistics and probability includes examples and exercises. Graduate students and researchers will find this text an excellent resource, requiring for mastery a solid foundation in probability theory, measure and integration, as well as some knowledge of stochastic processes and martingales. However, the material is presented so as to be accessible to a wider cross-disciplinary audience.