  In recent years several classical results in extremal graph theory have been
improved in a uniform way and their proofs have been simplified and
streamlined. These results include a new Erd\H{o}s-Stone-Bollob\'as theorem,
several stability theorems, several saturation results and bounds for the
number of graphs with large forbidden subgraphs. Another recent trend is the
expansion of spectral extremal graph theory, in which extremal properties of
graphs are studied by means of eigenvalues of various matrices. One particular
achievement in this area is the casting of the central results above in
spectral terms, often with additional enhancement. In addition, new, specific
spectral results were found that have no conventional analogs. All of the above
material is scattered throughout various journals, and since it may be of some
interest, the purpose of this survey is to present the best of these results in
a uniform, structured setting, together with some discussions of the
underpinning ideas.
