One of the most natural and important questions in statistical learning is how well a distribution can be approximated from its samples. Surprisingly, this question has so far been resolved for only a few approximation measures, for example the KL-divergence, and even then the answer is ad hoc and not well understood. We resolve the question for three more important approximation measures, $\ell_1$, $\ell_2^2$, and $\chi^2$, and if the probabilities are bounded away from zero, we resolve the question for all smooth $f$-divergence approximation measures, thereby providing a coherent understanding of the rate at which a distribution can be approximated from its samples.