Embedding graphs into larger graphs: results, methods, and problems

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which --- for some other reasons --- are very close to us. Some results discussed here got stronger emphasis, since they are connected to Lov\'asz (and sometimes to us).Comment: 153 pages, 15 figures, 3 tables. Almost final version of the survey for Building Bridges II (B\'ar\'any et al eds.), Laszlo Lovasz 70th Birthda