In this paper general conditions are given for the validity of multivariate Edgeworth expansions for a sequence of random vectors. The main difference between the author's approach and the classical one [see, e.g., the monograph by \textit{R. N. Bhattacharya} and \textit{R. R. Rao}, ''Normal approximation and asymptotic expansions.'' (1976; Zbl 0331.41023)] is that different rates of convergence are allowed in different directions (i.e. the variance may tend to infinity at different rates in different directions). Well-known results cannot be applied directly because, as a consequence, also the treatment of the characteristic function must be different in each direction. Both asymptotic expansions for distribution functions as well as uniform local asymptotic expansions for densities of a sequence of random vectors are considered. The verification of the assumptions of the author's main results (Theorem 3.1, Lemma 3.2 and Corollary 3.3) is very difficult in general, but can be carried out for the important special case of sums of independent random variables. Also sums of a certain type of dependent random variables as well as analogous results for the lattice case are briefly discussed.