<p>The article is a review of our results on the compactness theorems for the families of solutions of the Beltrami equation and the corresponding Dirichlet problem. The review includes the following results: 1) theorems on the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation in some Jordan domain whose characteristics have a compact support and satisfy certain constraints of an integral nature; 2) theorems on the compact classes of solutions of the Dirichlet problem for the Beltrami equation with integral constraints, which is considered in some Jordan domain; 3) results on the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation, whose characteristics have a compact support and satisfy certain constraints of set-theoretic type; 4) results on the compact classes of solutions of the Dirichlet problem for the Beltrami equation, which are considered in some Jordan domain and whose characteristics satisfy certain set-theoretic-type constraints; 5) results on the compactness of solutions of the Dirichlet problem in a simply connected domain, which are obtained in terms of prime ends in the case when the maximal dilations of these solutions satisfy certain integral constraints; 6) theorems on the existence of homeomorphic <i>ACL</i>-solutions of the quasilinear Beltrami equation with two characteristics under certain conditions on complex coefficients; 7) theorems on the existence of continuous <i>ACL</i>-solutions of the Beltrami equation, which are logarithmically Holder in a given domain; and 8) results on the existence of continuous solutions of the Beltrami equations with two characteristics, which satisfy a certain condition on the dilation of inverse mappings, and the solutions themselves satisfy the condition of hydrodynamic normalization in the vicinity of infinity. The vast majority of the results were included in the Ph.D. thesis of O.P.D.</p>