This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of 0(m0) operations, where m0 is the number of edges in the initial graph, the expected time for p updates is O(p log3 n) (Throughout the paper the logarithms are base 2.) for connectivity and bipartiteness. The worst-case time for one query is O(log n/log log n). For the k-edge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p.pow(log(n),3)) and the expected time for q queries is O(p_k.pow(log(n),3)). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O(p_k.pow(log(n),3)). This implies an algorithm to maintain a 1 + e- approximation of the minimum spanning tree in expected time O((p.pow(log(n),3).log U)/e) for p updates, where the weights of the edges are between 1 and U.