Levenshtein introduced the problem of constructing k-deletion correcting codes in 1966, proved that the optimal redundancy of those codes is O(k log N), and proposed an optimal redundancy single-deletion correcting code (using the so-called VT construction). However, the problem of constructing optimal redundancy k-deletion correcting codes remained open. Our key contribution is a solution to this long-standing open problem. We present a k-deletion correcting code that has redundancy 8k log n + o(log n) and encoding/decoding algorithms of complexity O(n  2k+1  ) for constant k.