  In this paper, we investigate the boundary behavior of solutions of
divergence-form operators with an elliptic symmetric part and a $BMO$
anti-symmetric part. Our results will hold in non-tangentially accessible (NTA)
domains; these general domains were introduced by Jerison and Kenig and include
the class of Lipschitz domains. We establish the H\"older continuity of the
solutions at the boundary, existence of elliptic measures $\omega_L$ associated
to such operators, and the well-posedness of the continuous Dirichlet problem
as well as the $L^p(d\omega)$ Dirichlet problem in NTA domains. The equivalence
in the $L^p$ norm of the square function and the non-tangential maximal
function under certain conditions remains valid. When specialized to Lipschitz
domains, it is then possible to extend, to these operators, various criteria
for determining mutual absolute continuity of elliptic measure with surface
measure.
