This book shows how to design and analyze algorithms that make decisions in the face of uncertainty about the future. The first of two volumes covers the foundations of competitive analysis alongside a broad selection of classical and modern problems, including Ski Rental, Paging, k-Server, Metrical Task Systems, Bipartite Matching and Facility Location. It also explores connections to online learning, regret minimization, mechanism design and social choice, reflecting the field's growing breadth and impact. Each chapter is enriched with extended historical notes situating results in their research context, selected exercises ranging from routine to research level, and open problems pointing toward active frontiers. The text functions equally well as a coursebook and as a research reference, ideal for advanced undergraduate students, graduate students and active researchers in computer science who want to bring their understanding of online algorithms to the current research frontier.