Revision with unchanged content. Many random graph processes exhibit a phase transition, where the compo-nent structure of the graph changes radically caused by the addition of relatively few random edges. Before the phase transition the graph consists, with high probability, of many small components, while after the phase transition it contains, with high probability, a unique component of linear size, called the giant component, which is much larger than every other component in the graph. This book treats the phase transition and the emergence of the giant component in three different random graph models. It presents several techniques that are useful for studying this type of problem, including generating functions, branching processes and differential equations. The book is aimed at mathematicians interested in random graphs in general, and in the phase transition and the giant component in particular.