Invitation to intersection problems for finite sets

Extremal set theory is dealing with families, FF of subsets of an n  -element set. The usual problem is to determine or estimate the maximum possible size of FF, supposing that FF satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any r   subsets in FF have at least t elements in common, all the sizes of pairwise intersections belong to a fixed set, L of natural numbers, there are no s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bounds.