The period since the publication of the first edition of this book has seen the theory of random graphs go from strength to strength. Indeed, its appearance happened to coincide with a watershed in the subject; the emergence in the subsequent few years of significant new ideas and tools, perhaps most notably concentration methods, has had a major impact. It could be argued that the subject is now qualitatively different, insofar as results that would previously have been inaccessible are now regarded as routine. Several longstanding issues have been resolved, including the value of the chromatic number of a random graph Gn,p, the existence of Hamilton cycles in random cubic graphs, and precise bounds on certain Ramsey numbers. It remains the case, though, that most of the material in the first edition of the book is vital for gaining an insight into the theory of random graphs.

It would be impossible, in a single volume, to prove all the substantial new results that we would wish to, so we have chosen instead to give brief descriptions and to sketch a number of proofs.