Let L be an arbitrary second-order, homogeneous, elliptic system, with constant complex coefficients. We obtain that the Dirichlet problem in the upper-half space with boundary data in BMO (the space of bounded mean oscillation functions) is well-posed when the gradient of the solution satisfies a Carleson measure condition. Analogously, we consider boundary data in VMO (the space of vanishing mean oscillation functions) and show that the corresponding problem is well-posed when the gradient of the solution satisfies a vanishing Carleson measure condition. We also establish Fatou type theorems guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems. In concert, these results imply that the spaces BMO and VMO can be characterized as the collection of nontangential pointwise traces of null-solutions to elliptic systems satisfying the mentioned Carleson measure conditions. Our results extend the work of Fabes, Johnson, and Neri, who considered the BMO Dirichlet problem for the Laplacian, and apply to a much larger class of PDE's, including the Lamé system of elasticity, among others. Joint work with D. Mitrea, I. Mitrea and M. Mitrea.