The bootstrapped size and power properties of six long memory tests the modified R/S, KPSS, V/S, GPH, Robinson's H and the recently proposed Sk tests are investigated. Even in samples of size 100, the moving block bootstrap controls the empirical size of the tests in the DGPs examined. The H test appears to be the most powerful. Moreover, compared with asymptotic tests, the bootstrap tests suffer little loss of power against fractionally integrated processes in samples with 250 or more observations. This is true both for distributions with heavy tails and with stochastic volatility.