The topic of the present paper is the following question. Let G be a locally finite group admitting an automorphism f of finite order such that the centralizer CG(f) satisfies certain finiteness conditions. What impact does this have on the structure of the group G? Equivalently, one can ask the same question when f is an element of G. Sometimes the impact is quite strong and the paper is a survey of results illustrating this phenomenon. In particular, we concentrate on results where G is shown to have a large nilpotent or soluble subgroup. Naturally, in each case the result depends on the order of the automorphism f and kind of conditions imposed on CG(f). We shall be considering mostly the classical finiteness conditions such as CG(f) being finite, Chernikov, and of finite rank, respectively. It is not a purpose of the paper to survey numerous results on automorphisms of finite groups. In particular, among important topics that are left out of the present discussion are ``p-automorphisms of p-groups\" (see [36]) and ``length problems\" (see [73]). However, in some situations (like, for example, when CG(f) is finite) problems on infinite groups quickly reduce to finite groups and in those cases working with finite groups is very natural. A separate section of the paper is devoted to the case when f is of order two, a prime, four, or other, respectively. However, before anything else we address in Section 1 the following related question. Given a periodic group G with an automorphism f, what additional assumptions on G and CG(f) ensure that G is locally finite?

The topic of the present paper is the following question. Let G be a locally finite group admitting an automorphism f of finite order such that the centralizer CG(f) satisfies certain finiteness conditions. What impact does this have on the structure of the group G? Equivalently, one can ask the same question when f is an element of G. Sometimes the impact is quite strong and the paper is a survey of results illustrating this phenomenon. In particular, we concentrate on results where G is shown to have a large nilpotent or soluble subgroup. Naturally, in each case the result depends on the order of the automorphism f and kind of conditions imposed on CG(f). We shall be considering mostly the classical finiteness conditions such as CG(f) being finite, Chernikov, and of finite rank, respectively. It is not a purpose of the paper to survey numerous results on automorphisms of finite groups. In particular, among important topics that are left out of the present discussion are ``p-automorphisms of p-groups\" (see [36]) and ``length problems\" (see [73]). However, in some situations (like, for example, when CG(f) is finite) problems on infinite groups quickly reduce to finite groups and in those cases working with finite groups is very natural. A separate section of the paper is devoted to the case when f is of order two, a prime, four, or other, respectively. However, before anything else we address in Section 1 the following related question. Given a periodic group G with an automorphism f, what additional assumptions on G and CG(f) ensure that G is locally finite?