This paper examines the random walk hypothesis (RWH) and the martingale difference hypothesis (MDH) for the Australian dollar and five Asian emerging currencies relative to three benchmark currencies. We use Wright’s (2000) non-parametric procedure to test the RWH and Kuan and Lee’s (2004) procedure to test the MDH. The results of Wright’s tests and Kuan and Lee’s test are adjusted for size distortion. The RWH is rejected for all currencies before and after the Asian crisis. The results of Kuan and Lee’s test are consistent with the fact that the RWH is more stringent than the MDH. For the three testing periods, the MDH fails to reject the AUD. For all other currencies the MDH is rejected at least for one benchmark over two periods, indicating that the market efficiency in these markets have not significantly improved under the floating rate systems following the Asian financial crisis.