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We prove that for any homogeneous, second order, constant complex coefficientelliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ withboundary data in BMO is well-posed in the class of functions $u$ with$d\mu_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. Weestablish a Fatou type theorem guaranteeing the existence of the pointwisenontangential boundary trace for smooth null-solutions $u$ of such systemssatisfying the said Carleson measure condition. These imply that BMO can becharacterized as the collection of nontangential pointwise traces of smoothnull-solutions $u$ to the elliptic system $L$ with the property that $\mu_u$ isa Carleson measure.  We establish a regularity result for the BMO-Dirichlet problem in theupper-half space: the nontangential pointwise trace of any given smoothnull-solutions of $L$ satisfying the above Carleson measure condition belongsto Sarason's space VMO if and only if $\mu_u$ satsifies a vanishing Carlesonmeasure condition. Moreover, we obtain the well-posedness of the Dirichletproblems when the boundary data are prescribed in Morrey-Campanato andsolutions are required to satisfy a vanishing Carleson measure condition offractional order.  As a consequence, we characterize the space VMO as the closure in BMO ofclasses of smooth functions contained in BMO within which uniform continuitymay be suitably quantified (such as the class of smooth functions satisfying aH\"older or Lipschitz condition), improving Sarason's classical resultdescribing VMO as the closure in BMO of the space of uniformly continuousfunctions with bounded mean oscillations.  Finally, we show that any Calder\'on-Zygmund operator $T$ satisfying $T(1)=0$extends as a bounded linear operator in $\mathrm{VMO}$, and characterize themembership to $\mathrm{VMO}$ via the action of various classes of singularintegral operators.

The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO