We develop the central idea of emergence in the uniform random graph models of this book. We illustrate this with old unsolved problems connected with circle and sphere packings. We introduce notions of entropy, phase space, phases, phase transitions, and order parameters imported from statistical mechanics. We then adapt these notions to define uniform random graph models and in particular the edge-triangle model, which uses a uniform distribution on graphs with given densities of edges and triangles. We argue that the techniques of this book are broadly applicable to other optimization problems, for instance Plateau’s problem and Optimal Transport. We close with a summary of the results of the book, introducing a conjectured phase diagram for the edge-triangle model and indicating which features of that diagram are rigorously proven in later chapters.