A {\em local graph partitioning algorithm} finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number $n$ of vertices in $G$. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the {\em volume-biased evolving set process}, which is a Markov chain on sets of vertices. We prove that for any set of vertices $A$ that has conductance at most $φ$, for at least half of the starting vertices in $A$ our algorithm will output (with probability at least half), a set of conductance $O(φ^{1/2} \log^{1/2} n)$. We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is $O(φ^{-1/2} polylog(n))$. In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of $O(φ^{-1} polylog(n))$ between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of $φ$, the resulting algorithm has complexity $O((m+nφ^{-1/2}) polylog(n))$ and returns a cut with conductance $O(φ^{1/2} \log^{1/2} n)$ and volume at least $v_φ/2$, where $v_φ$ is the largest volume of any set with conductance at most $φ$.