Summary: The general stability theory of the synchronized motions of the coupled-oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state \(\Phi_{unif}\). When the coupling strength is weakened, the coupled system may exhibit several types of non-synchronized motion. In particular, if \(\Phi_{unif}\) is chaotic, we always get a transition from chaotic \(\Phi_{unif}\) to a certain non-uniform state and finally the non-uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.