Asbtract:  We first consider robust permutation tests for a location shift in the two sample case based on estimating equations, comparing the test statistics based on a score function and M-estimates. We obtain a form for these tests so that the exact tests may be carried out using the same algorithms as used for permutation tests based on the mean and we obtain some numerical results indicating the increase in power when using robust tests. We consider asymptotics for these test statistics from which we obtain saddlepoint approximations for tail probabilities which maintain second order relative errors and enable us to consider Bahadur slopes of the tests based on the two statistics. We show that they have different Bahadur slopes with neither exceeding the other over the whole range. We also report some simulations confirming these large deviation results and indicating the accuracy of the saddlepoint approximations. We consider extensions to the k sample robust permutation test and a two sample multivariate test based on estimating equations. In both cases we need to choose a test statistic for such a multi-parameter test. Rather than the usual sum of squares we use a natural convex test statistic which permits an accurate saddlepoint approximation. We give the saddlepoint approximations and obtain some numerical results illustrating their accuracy.