This book is a mathematical cross-section of the fundamentals of modern Machine Learning. The main ideas behind the three main paradigms of ML (Supervised, Unsupervised and Reinforcement Learning) are analysed from the perspective of the underlying mathematical theory that justifies their successes in state-of-the-art applications. All the results are rigorously proven, and references are given where appropriate, to help build up a bag of theoretical tools for the ML practitioner.