  In this paper we introduce a unifying approach to the generalized Tur\'an
problem and supersaturation results in graph theory. The
supersaturation-extremal function $satex(n, F : m, G)$ is the least number of
copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least
$m$ copies of $F$ as a subgraph. We present a survey, discuss previously known
results and obtain several new ones focusing mainly on proof methods, extremal
structure and phase transition phenomena. Finally we point out some relation
with extremal questions concerning hypergraphs, particularly Berge-type
results.
