This book provides a concise, self-contained introduction to nonsmooth convex analysis and optimization. It develops the core ideas of nonsmooth analysis, subdifferential calculus, and monotone operator theory that form the basis of proximal algorithms for convex optimization. The presentation is rigorous yet accessible, and is aimed at graduate and advanced undergraduate students, as well as researchers in operations research, statistics, signal processing, control and machine learning. The only prerequisites are linear algebra and basic real analysis, and most of the material is developed from first principles. Numerous examples and exercises are included to support learning and self-study, and the material is suitable for a one-semester course.