The Lyapunov functions method is a powerful stability analysis tool of complex dynamic systems. In this work, it is revised for stability analysis under uncertain conditions to augment the arsenal of existing Lyapunov functions by admitting their dependence on external disturbances affecting the system. To this end, robust stability is specified for nonlinear systems affected by uncertain disturbances. Their analysis is then developed in terms of disturbance-dependent Lyapunov functions which are conceptualized for the systems in question. It is shown that robust stability is guaranteed in the presence of such a Lyapunov function. The effectiveness of the proposed analysis tools is illustrated by a nontrivial example.