Сharacterizations of Layer-Finite Groups and Their Extensions

In the paper we review the results characterizing layer-finite groups and almost layer-finite groups in the class of all groups. We also characterize the groups with layer-finite periodic parts, the groups with almost layer-finite periodic parts and the groups with generalized Chernikov periodic parts. Theorems describing the properties of quasi-layer-finite groups and simple quasi-layer-finite groups are provided.