In this paper we report on some recent results about maximal surfaces in a Lorentzian product space of the form M 2 × R 1 , where M 2 is a connected Riemannian surface and M 2 × R 1 is endowed with the Lorentzian metric , = , M − dt 2 . In particular, if the Gaussian curvature of M is non-negative, we establish new Calabi-Bernstein results for complete maximal surfaces immersed into M 2 × R 1 and for entire maximal graphs over a complete surface M . We also construct counterexamples which show

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... hat our Calabi-Bernstein results are no longer true without the hypothesis K M ≥ 0. Finally, we introduce two local approaches to our global results. We do not provide here with detailed proofs of our results. For further details, we refer the reader to the original papers Ref. 1-3.