  This article is a survey on ind-varieties and ind-groups introduced by
Shafarevich in 1965, with a special emphasis on automorphism groups of affine
varieties and actions of ind-groups on ind-varieties. We give precise
definitions and complete proofs, including several known results. The survey
contains many examples and also some questions which came up during our work on
the subject.
  Among the new results we show that for an affine variety X the automorphism
group Aut(X) is always locally closed in the ind-semigroup End(X) of all
endomorphisms, and we give an example of a strict closed subgroup of a
connected ind-group which has the same Lie algebra, based on the work of
Shestakov-Umirbaev on the existence of non-tame automorphisms of affine
3-space.
