This chapter introduces the foundations and motivations of federated learning (FL), a distributed machine learning approach that enables model training directly on devices where data is generated. It emphasizes the relevance of FL in modern settings where decentralized data-such as from smartphones, wearables, and weather stations-exhibits structural relationships and privacy constraints. FL addresses challenges associated with data centralization by leveraging local computation and network-based optimization. Key drivers of FL include privacy, robustness to failures and attacks, parallelization, and personalization. Structurally, the chapter begins by motivating FL through real-world examples and then outlines core mathematical tools such as Euclidean spaces, gradient-based methods, and fixed-point iterations. Finally, it offers a roadmap of the book, organized into three parts: a review of machine learning principles, a formal treatment of FL models and methods, and a discussion of ethical and practical requirements for trustworthy AI. The chapter closes with exercises to reinforce foundational concepts in linear algebra and optimization.