The text provides an introduction to mathematics and the application possibilities of Monte Carlo methods, consistently using the language of stochastic processes. The reader will learn the fundamental principles and essential characteristics of these methods, enabling them to utilize this important algorithmic tool and statistically interpret the results. Through selected questions, the reader will also be introduced to current research issues in this field. Topics covered include direct simulation, methods for simulating distributions and stochastic processes, variance reduction, as well as introductory-level Markov Chain Monte Carlo methods and high-dimensional integration. Application examples from particle physics and financial and actuarial mathematics are presented.