We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph-which undergoes both edge insertions and deletions-in polylogarithmic time. Our algorithm is randomized and, per update, takes O(log ² Δ log ² n) expected time. Furthermore, the algorithm can be adjusted to have O(log ² Δ log ⁴ n) worst-case update-time with high probability. Here, n denotes the number of vertices and Δ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with O(m ^3/4 ) update-time (and thus broke the natural Ω(m) barrier) where m denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had Õ(min{√n, m ^1/3 }) update-time [Assadi et al. SODA'19]. Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.