Caching refers to the act of replicating information at a faster (or closer) medium with the purpose of improving performance. This deceptively simple idea has given rise to some of the hardest optimization problems in the fields of computer systems, networking, and the Internet; many of which remain unsolved several years after their conception. While a wealth of research contributions exists from the topics of memory systems, data centers, Internet traffic, CDNs, and recently wireless networks, the literature is dispersed and overlapping at times. In this monograph, the authors focus on the fundamental underlying mathematical models, into a powerful framework for performing optimization of caching systems. In doing so they the present the reader with a solid background for the anticipated explosion in caching research, and provide a didactic view into how engineers have managed to infuse mathematical models into the study of caching over the last 40 years. Written by leading researchers from academia and industry, this monograph provides students, researchers and practicing engineers with a concise introduction to challenges and solutions for implementing caching in modern computing systems.