This paper presents optimal tests for parameter instability in the GMM framework. The new tests include tests that are optimal for both one-sided and two-sided alternatives. One of the optimal tests for two-sided alternatives is the GMM generalization of the test presented in Andrews and Ploberger (1994) for the likelihood framework. The new tests include a class of optimal tests that direct the test's power to specific locations in the sample. One of these optimal tests has the attractive feature of a normal distribution under the null hypothesis.