We study Nyström type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling and high probability estimates are considered. In particular, we prove that these approaches can achieve optimal learning bounds, provided the subsampling level is suitably chosen. These results suggest a simple incremental variant of Nyström kernel ridge regression, where the subsampling level controls at the same time regularization and computations. Extensive experimental analysis shows that the considered approach achieves state of the art performances on benchmark large scale datasets.