Let M 2 be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in M 2 × R.First, using a characterization of δ-parabolicity we prove that under additional conditions on M, an embedded minimal surface with bounded gaussian curvature is proper.The second theorem states that under some conditions on M, if Σ is a properly immersed minimal surface with finite topology and one end in M × R, which is transverse to a slice M × {t} except at a finite number of points, and such that Σ∩(M×{t}) contains a finite number of components, then Σ is parabolic.In the last result, we assume some conditions on M and prove that if a minimal surface in M × R has height controlled by a logarithmic function, then it is parabolic and has a finite number of ends.