In the present work the Carleman-Bers-Vekua equations on the complex plane with the regular coefficients are considered. We have introduced special classes of solutions of these equations, which we call generalized meromorphic functions, and we study them both from the point of view of the pure theory of functions, as well as from the point of view of the analysis of boundary value problems of the theory of functions. Sufficiently important information about the behavior (hence the structure) of the generalized meromorphic function in the neighborhood of the point at infinity has been determined. Based on these results, it is possible to correctly pose natural boundary value problems for generalized meromorphic functions and carry out their (in some sense) complete analysis.