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In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical one-way model, our bound extends the well known upper bound of Kremer, Nisan and Ron [I. Kremer, N. Nisan, D. Ron, On randomized one-round communication complexity, in: Proceedings of The 27th ACM Symposium on Theory of Computing, STOC, 1995, pp. 596-605] to include non-product distributions. Let @e@?(0,1/2) be a constant. We show that for a boolean function f:XxY-{0,1} and a non-product distribution @m on XxY, D"@e^1^,^@m(f)=O((I(X:Y)+1)@?VC(f)), where D"@e^1^,^@m(f) represents the one-way distributional communication complexity of f with error at most @e under @m; VC(f) represents the Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual information, under @m, between the random inputs of the two parties. For a non-boolean function f:XxY-{1,...,k} (k=2 an integer), we show a similar upper bound on D"@e^1^,^@m(f) in terms of k,I(X:Y) and the pseudo-dimension of f^'=deffk, a generalization of the VC-dimension for non-boolean functions. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f, in terms of the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f. We show for a non-boolean total function f:XxY-Z and a product distribution @m on XxY, Q"@e"^"3"/"8^1^,^@m(f)=@W(rec"@e^1^,^@m(f)), where Q"@e"^"3"/"8^1^,^@m(f) represents the quantum one-way distributional communication complexity of f with error at most @e^3/8 under @m and rec"@e^1^,^@m(f) represents the one-way rectangle bound of f with error at most @e under @m. Similarly for a non-boolean partial function f:XxY-Z@?{*} and a product distribution @m on XxY, we show, Q"@e"^"6"/"("2"@?"1"5"^"4")^1^,^@m(f)=@W(rec"@e^1^,^@m(f)).