This article is about tests for serial, spatial, or other correlation in e, the error vector in the linear model Y = Xβ + e. The errors need not come from a particular parametric distribution such as the normal. P values are computed using permutations of the regression residuals. In the special cases that Y = e or X is a vector of Is, the tests are valid and consistent. In general, regression residuals are not exchangeable, so the tests are only approximate. It is shown, however, that under mild conditions, the permutation tests are asymptotically valid and consistent. The test statistics are asymptotically normal. In the serial case, the tests are closely related to the Durbin-Watson (DW) test. The spatial case is illustrated with a test based on a triangulation of observation locations. In addition to providing a spatial analog to the DW test, the permutation tests circumvent two of the DW test's difficulties: dependency on the normality of errors and the need to accommodate “indifference zones.”