Recnrsive Diagonal Torus, or RDT consisting of recursively structured tori is an interconnection network tbr massively parMlel computers. By adding remote links to the diagonal directions of the torus network recursively, the diameter can be reduced within log2N with smaller number of links than that of hypercube. For an interconnection network for massively parallel computers, a routing algorithm which can bypass a faulty or congested node are essential. Although the conventional vector routing is a simple and near-optimM method, it can only use a deterministic path. In this paper, adaptive routing algorithms on RDT are proposed and discussed. The first algorithm is based on Duato's necessary and sufficient condition. With this method virtual channels are effectively used while paths with redundant routing steps are prohibited. Another algorithm based on the turn model is proposed. By prohibiting certain turns on RDT, it permits paths with additional hops. Both algorithms are proved to be deadlock free.