  In this paper we establish new Calabi-Bernstein results for maximal surfaces
immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$,
where $M^2$ is a connected Riemannian surface and $M^2\times\mathbb{R}_1$ is
endowed with the Lorentzian metric $<,>=<,>_{M}-dt^2$. In particular, when $M$
is a Riemannian surface with non-negative Gaussian curvature $K_M$, we prove
that any complete maximal surface in $M^2\times\mathbb{R}_1$ must be totally
geodesic. Besides, if $M$ is non-flat we conclude that it must be a slice
$M\times\{t_0\}$, $t_0\in\mathbb{R}$ (here by "complete" it is meant, as usual,
that the induced Riemannian metric on the maximal surface from the ambient
Lorentzian metric is complete). We prove that the same happens if the maximal
surface is complete with respect to the metric induced from the Riemannian
product $M^2\times\mathbb{R}$. This allows us to give also a non-parametric
version of the Calabi-Bernstein theorem for entire maximal graphs in
$M^2\times\mathbb{R}_1$, under the same assumptions on $K_M$. Moreover, we also
construct counterexamples which show that our Calabi-Bernstein results are no
longer true without the hypothesis $K_M\geq 0$. These examples are constructed
via a duality result between minimal and maximal graphs.
