Minimal surfaces in M 2× R

We study the geometry and topology of properly embedded minimal surfaces in M, where M is a Riemannian surface. When M is a round sphere, we give examples of all genus and we prove such minimal surfaces have exactly two ends or equal M×\, for some real t. When M has non-negative curvature, we study the conformal type of minimal surfaces in M, and we prove half-space theorems. When M is the hyperbolic plane, we obtain a Jenkins-Serrin type theorem.