  We study the minimal surface equation in the Heisenberg space, Nil_3. A
geometric proof of non existence of minimal graphs over non convex, bounded and
unbounded domains is achieved (our proof holds in the Euclidean space as well).
We solve the Dirichlet problem for the minimal surface equation over bounded
and unbounded convex domains, taking bounded, piecewise continuous boundary
value. We are able to construct a Scherk type minimal surface and we use it as
a barrier to construct non trivial minimal graphs over a wedge of angle between
\pi/2 ,and \pi taking non negative continuous boundary data, having at least
quadratic growth. In the case of an half- plane, we are also able to give
solutions (with either linear or quadratic growth), provided some geometric
hypothesis on the boundary data. Finally, some open problem arising from our
work, are posed.
