Matrix completion and low rank matrix recovery is a technique by which a low rank matrix is measured either by sampling entries of the matrix or through matrix products. Rank $r$ matrices of size $m \times n$ have $r(m+n-r)$ degrees of freedom, requiring at least the same number of entries to be known. Computationally efficient algorithms, such as semidefinite programming, have been shown to be able to recover a rank $r$ matrix from a modest multiple of the number of degrees of freedom. In this talk we present new nonconvex algorithms for matrix completion and present empirical evidence that these algorithms are able to recover low rank matrices from a multiple of the oracle rate where the multiple is observed to converge towards one, giving the optimal rate.