This is a masters-level overview of the mathematical concepts needed to fully grasp the art of derivatives pricing, and a must-have for anyone considering a career in quantitative finance in industry or academia. Starting from the foundations of probability, this textbook allows students with limited technical background to build a solid knowledge of the most important principles. It offers a unique compromise between intuition and mathematics, even when discussing abstract ideas such as change of measure. Mathematical concepts are introduced initially using toy examples, before moving on to examples of finance cases, both in discrete and continuous time. Throughout, numerical applications and simulations illuminate the analytical results. The end-of-chapter exercises test students' understanding, with solved exercises at the end of each part to aid self-study. Additional resources are available online, including slides, code and an interactive app.