This paper presents an algorithm for the fully dynamic biconnectivity problem whose running time is exponentially faster than all previously known solutions. It is the first dynamic algorithm that answers biconnectivity queries in time O(log2 n) in a n-node graph and can be updated after an edge insertion or deletion in polylogarithmic time. Our algorithm is a Las-Vegas style randomized algorithm with the update time amortized update time O(log4 n). Only recently the best deterministic result for this problem was improved to O(Vn log2 n). We also give the first fully dynamic and a novel deletions-only transitive closure (i.e. directed connectivity) algorithms. These are randomized Monte Carlo algorithms. Let n be the number of nodes in the graph and let m be the average number of edges in the graph during the whole update sequence: The fully dynamic algorithms achieve (1) query time O(n/log n) and update time O(mVn log2 n+n); or (2) query time O(n/log n) and update time O[n.pow(m,(u-1)/u)].pow2(log(n)) = O[n.pow(m,0.58).pow2(log(n)], where u is the exponent for boolean matrix multiplication (currently u = 2.38). The deletions-only algorithm answers queries in time O(n/log n). Its amortized update time is O(n log2 n).