We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved by previous efficient algorithms. Firstly, we study mixtures of $k$ distributions in $d$ dimensions, where the means of every pair of distributions are separated by at least $k^{\varepsilon}$. In the special case of spherical Gaussian mixtures, we give a $(dk)^{O(1/\varepsilon^2)}$-time algorithm that learns the means assuming separation at least $k^{\varepsilon}$, for any $\varepsilon > 0$. This is the first algorithm to improve on greedy ("single-linkage") and spectral clustering, breaking a long-standing barrier for efficient algorithms at separation $k^{1/4}$. We also study robust estimation. When an unknown $(1-\varepsilon)$-fraction of $X_1,\ldots,X_n$ are chosen from a sub-Gaussian distribution with mean $μ$ but the remaining points are chosen adversarially, we give an algorithm recovering $μ$ to error $\varepsilon^{1-1/t}$ in time $d^{O(t^2)}$, so long as sub-Gaussian-ness up to $O(t)$ moments can be certified by a Sum of Squares proof. This is the first polynomial-time algorithm with guarantees approaching the information-theoretic limit for non-Gaussian distributions. Previous algorithms could not achieve error better than $\varepsilon^{1/2}$. Both of these results are based on a unified technique. Inspired by recent algorithms of Diakonikolas et al. in robust statistics, we devise an SDP based on the Sum of Squares method for the following setting: given $X_1,\ldots,X_n \in \mathbb{R}^d$ for large $d$ and $n = poly(d)$ with the promise that a subset of $X_1,\ldots,X_n$ were sampled from a probability distribution with bounded moments, recover some information about that distribution.