Textbook in PDF format This book provides a broad introduction to algorithms for validating safety-critical systems. We cover a wide variety of topics related to validation, introducing the underlying mathematical problem formulations and the algorithms for solving them. Figures, examples, and exercises are provided to convey the intuition behind the various approaches. This book is intended for advanced undergraduates and graduate students, as well as professionals. It requires some mathematical maturity and assumes prior exposure to multivariable calculus, linear algebra, and probability concepts. Some review material is provided in the appendices. Disciplines where the book would be especially useful include mathematics, statistics, computer science, aerospace, electrical engineering, and operations research. Fundamental to this textbook are the algorithms, which are all implemented in the Julia programming language. We have found this language to be ideal for specifying algorithms in human-readable form. The priority in the design of the algorithmic implementations was interpretability rather than efficiency. Industrial applications, for example, may benefit from alternative implementations. Permission is granted, free of charge, to use the code snippets associated with this book, subject to the condition that the source of the code is acknowledged