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We give a randomized 2^{n+o(n)}-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic O(4^n)-time and O(2^n)-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2^(n/2) vectors from the discrete Gaussian distribution at any parameter in 2^(n+o(n)) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a 2^(n+o(n))-time algorithm that approximates the Closest Vector Problem to within a factor of 1.97. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2^(n/2) discrete Gaussian samples in just 2^(n/2+o(n)) time and space. Among other things, this implies a 2^(n/2+o(n))-time and space algorithm for 1.93-approximate decision SVP.