We develop the basic notions used throughout the book, providing precise definitions of graphons, Boltzmann and Shannon entropies, phases, and phase transitions, and give proofs of some essential properties. We present graphons from three perspectives, as limits of large graphs, as lists of subgraph densities, and as random processes for generating graphs. A Large Deviations Principle (LDP) relates the Shannon entropy of a graphon to the number of finite graphs approximated by the graphon. Using this LDP, we show that the Boltzmann entropy equals the maximal Shannon entropy of those graphons meeting certain integral constraints. If there is a unique graphon maximizing the Shannon entropy, then typical graphs in the ensemble are described by that graphon. This reduces most questions about phases in uniform random graph models to entropy maximization problems, with the bulk of the book devoted to such problems. To support this, we develop the theory of Lagrange multipliers for graphons, and in particular the "worth" of a column of a graphon, proving that all columns of an optimal graphon have the same worth.