This paper is devoted to the study of the H-function as defined by the Mellin-Barnes integral Hp,qm,n(z)=12πip,qm,n(s)zsds, where the function p,qm,n(s) is a certain ratio of products of the Gamma-functions with the argument s and the contour specially chosen. The conditions for the existence of Hp,qm,n(z) are discussed and explicit power and power-logarithmic series expansions of Hp,qm,n(z) near zero and infinity are given. The obtained results define more precisely the known results.