Monte Carlo methods are a family of algorithms with far-reaching applications across science, engineering, statistics, and applied mathematics. This textbook provides a concise mathematical introduction to these methods, emphasizing core principles in the design and analysis of such methods through a carefully chosen selection of topics.The text focuses on timeless ideas rather than rapidly changing implementations. Algorithms are introduced in a way that supports conceptual understanding and mathematical analysis, with clarity and intuition favored over technical refinements or ad-hoc heuristics. To help readers navigate the broad landscape of Monte Carlo methods, each algorithm is accompanied by a discussion of its strengths, limitations, practical relevance, and the types of problems for which it is most suitable. Each chapter concludes with an exercise section and bibliographic remarks that point readers toward recent developments and further directions. This is a comprehensive resource for students in the classroom and during independent study.