In this paper, we develop the theory of properly embedded minimal surfaces in , where is a closed orientable Riemannian surface. We construct many examples of different topology and geometry. We establish several global results. The first of these theorems states that examples of bounded curvature have linear area growth, and so, are quasiperiodic. We then apply this theorem to study and classify the stable examples. We prove the topological result that every example has a finite number of ends. We apply the recent theory of Colding and Minicozzi to prove that examples of finite topology have bounded curvature. Also we prove the topological unicity of the embedding of some of these surfaces.