Various interesting results on interpolation theory of entire functions with given growth conditions have been obtained by imposing conditions on multiplicity varieties and weights. All the results discussed in the literature are limited to the space of entire functions. In this paper, we shall extend and generalize the interpolation problem of entire functions to meromorphic functions. The analytic conditions sufficient and necessary for a given multiplicity variety to be interpolating for meromorphic functions with given growth conditions will be obtained. Moreover, purely geometric characterization of interpolating varieties will be given for slowly decreasing radial weights which enable us to determine whether or not a given multiplicity variety is an interpolating variety by direct calculation. when weights grow so rapidly as to allow infinite order functions in the considered space, the geometric conditions would become more delicate. For such weights p(z), we also find purely geometric sufficient as well as necessary conditions provided that log p(exp r) is convex. As corollaries of our results, one obtains the corresponding results for the interpolation of entire functions.