The subgroup growth of a group is the study of the number of its index n subgroups as a function of n. We will present some sharp estimates on this growth for lattices L in higher rank semi-simple Lie groups G. A surprising phenomenom is that this growth depends only on the root system of G and not on L. In the most general case, the results depend on the generalized Riemann Hyphothesis but a number of results can be also proved unconditionally.