  According to Paul Erd\H{o}s [Some notes on Tur\'an's mathematical work, J.
Approx. Theory 29 (1980), page 4] it was Paul Tur\'an who "created the area of
extremal problems in graph theory". However, without a doubt, Paul Erd\H{o}s
popularized extremal combinatorics, by his many contributions to the field, his
numerous questions and conjectures, and his influence on discrete
mathematicians in Hungary and all over the world. In fact, most of the early
contributions in this field can be traced back to Paul Erd\H{o}s, Paul Tur\'an,
as well as their collaborators and students. Paul Erd\H{o}s also established
the probabilistic method in discrete mathematics, and in collaboration with
Alfr\'ed R\'enyi, he started the systematic study of random graphs. We shall
survey recent developments at the interface of extremal combinatorics and
random graph theory.
