This review deals with structural and functional features of glycoside hydrolases, a widespread group of enzymes present in almost all living organisms. Their catalytic domains are grouped into 120 amino acid sequence-based families in the international classification of the carbohydrate-active enzymes (CAZy database). At a higher hierarchical level some of these families are combined in 14 clans. Enzymes of the same clan have common evolutionary origin of their genes and share the most important functional characteristics such as composition of the active center, anomeric configuration of cleaved glycosidic bonds, and molecular mechanism of the catalyzed reaction (either inverting, or retaining). There are now extensive data in the literature concerning the relationship between glycoside hydrolase families belonging to different clans and/or included in none of them, as well as information on phylogenetic protein relationship within particular families. Summarizing these data allows us to propose a multilevel hierarchical classification of glycoside hydrolases and their homologs. It is shown that almost the whole variety of the enzyme catalytic domains can be brought into six main folds, large groups of proteins having the same three-dimensional structure and the supposed common evolutionary origin.