We present a randomized Local Computation Algorithm (LCA) with query complexity $poly(Δ) \cdot \log n$ for the Maximal Independent Set (MIS) problem. That is, the algorithm determines whether each node is in the computed MIS or not using $poly(Δ) \cdot \log n$ queries to the adjacency lists of the graph, with high probability, and this can be done for different nodes simultaneously and independently. Here $Δ$ and $n$ denote the maximum degree and the number of nodes. This algorithm resolves a key open problem in the study of local computations and sublinear algorithms (attributed to Rubinfeld).