  This is a survey on the automorphism groups in various classes of affine
algebraic surfaces and the algebraic group actions on such surfaces. Being
infinite-dimensional, these automorphism groups share some important features
of algebraic groups. At the same time, they can be studied from the viewpoint
of the combinatorial group theory, so we put a special accent on
group-theoretical aspects (ind-groups, amalgams, etc.). We provide different
approaches to classification, prove certain new results, and attract attention
to several open problems.
