We show that any mean-periodic function f can be represented in terms of exponential-polynomial solutions of the same convolution equation f satisfies, i.e., u ∗ f = 0 ( μ ∈ E ′ ( ℝ n ) ) . This extends to n -variables the work of L . Schwartz on mean-periodicity and also extends L . Ehrenpreis' work on partial differential equations with constant coefficients to arbitrary convolutors. We also answer a number of open questions about mean-periodic functions of one variable. The basic ingredient is our work on interpolation by entire functions in one and several complex variables.