Abstract: This work is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics (as well as those propositional logics in the semantics of which theories of relations play an essential r\^ole). This work has a survey character, too. The most frequently used algebras like cylindric--, relation--, polyadic--, and quasi--polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some directions we tried to give an overview of the most recent results and research trends, too.