This work explores Lyapunov characterizations of the input-output-to-state stability( IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of norm-estimators, and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.