  For half a century, Mackey and Green functors have been successfully used to
model the induction and restriction maps which are ubiquitous in the
representation theory of finite groups. In the examples, the latter maps are
typically distilled in some way from induction and restriction functors between
additive categories. This naturally leads to the notions of Mackey and Green
2-functors and to axioms which better capture this underlying layer of
information. Such structures have been used in algebra, geometry and topology
for a long time. We survey examples and applications of this young (yet
arguably overdue) theory.
