Abstract: I will discuss several results on the geometry of surfaces immersed in R^3 with small or bounded L^2 norm of |A|. For instance, we prove that if the L^2 norm of |A| and the L^p norm of the mean curvature, H, for p > 2, are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded L^2 norm of |A|, not necessarily small, then such a disk is graphical away from its boundary, provided that the L^p norm of H is sufficiently small, p > 2. These results are joint work with T. Bourni and they are related to previous work of Schoen-Simon and Colding-Minicozzi.