We develop two methods for the following fundamental statistical task: given an $ε$-corrupted set of $n$ samples from a $d$-dimensional sub-Gaussian distribution, return an approximate top eigenvector of the covariance matrix. Our first robust PCA algorithm runs in polynomial time, returns a $1 - O(ε\logε^{-1})$-approximate top eigenvector, and is based on a simple iterative filtering approach. Our second, which attains a slightly worse approximation factor, runs in nearly-linear time and sample complexity under a mild spectral gap assumption. These are the first polynomial-time algorithms yielding non-trivial information about the covariance of a corrupted sub-Gaussian distribution without requiring additional algebraic structure of moments. As a key technical tool, we develop the first width-independent solvers for Schatten-$p$ norm packing semidefinite programs, giving a $(1 + ε)$-approximate solution in $O(p\log(\tfrac{nd}ε)ε^{-1})$ input-sparsity time iterations (where $n$, $d$ are problem dimensions).