The spectral transform method is a standard numerical technique for solving par- tial differential equations on a sphere and is widely used in atmospheric circulation models. Re- cent research has identified several promising algorithms for implementing this method on mas- sively parallel computers; however, no detailed comparison of the different algorithms has previ- ously been attempted. In this paper, we describe these different parallel algorithms and report on computational experiments that we have conducted to evaluate their efficiency on parallel com- puters. The experiments used a testbed code that solves the nonlinear shallow water equations on a sphere; considerable care was taken to ensure that the experiments provide a fair compar- ison of the different algorithms and that the results are relevant to global models. We focus on hypercube- and mesh-connected multicomputers with cut-through routing, such as the Intel iPSC/860, DELTA, and Paragon, and the nCUBE/2, but we also indicate how the results extend to other parallel computer architectures. The results of this study are relevant not only to the spectral transform method but also to multidimensional fast Fourier transforms (FFTs) and other parallel transforms.