  In this paper we prove a general and sharp Asymptotic Theorem for minimal
surfaces in $H^2\times R$. As a consequence, we prove that there is no properly
immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve
homologous to zero in the asymptotic boundary of $ H^2\times R,$ say
$\partial_\infty H^2\times R$, such that $C$ is contained in a slab between two
horizontal circles of $\partial_\infty H^2\times R$ with width equal to $\pi.$
  We construct minimal vertical graphs in $H^2\times R$ over certain unbounded
admissible domains taking certain prescribed finite boundary data and certain
prescribed asymptotic boundary data. Our admissible unbounded domains $\Om$ in
$H^2\times \{0\}$ are non necessarily convex and non necessarily bounded by
convex arcs; each component of its boundary is properly embedded with zero, one
or two points on its asymptotic boundary, satisfying a further geometric
condition.
