Suppose each node and each edge of a network is independently faulty with probability at most p and q respectively, where 0 < p, q< 1 are ar- bitrary constants independent of the size of the network. For each fixed integer d≥ 2, we con- struct a network with O(N) nodes and with de- gree O(log log N) such that, after removing all the faulty nodes and edges, it still contains the N- node d-dimensional N1/d x ... x N1/d torus, and hence the mesh of the same size, with probability 1-N-(loglog N). This is derived as a consequence of a simple constant-degree construction which tol- erates random faults where the failure probability of each node is O(log- N). We also give a sim- ple constant-degree construction with O(N) nodes that tolerates O(N(1-2-4)/d) worst case faults.