This paper deals with the characterization of all nonlinear stabilizing controllers. Observer-based kernel representations are introduced to avoid the difficulty which occurs in constructing the state-space realizations of the parametrization of input-output operators. The parametrization of all stabilizing controllers for a given plant and a class of stabilizing plant-controller pairs are derived by employing observer-based kernel representations. Furthermore, the relationship between the results via kernel representation approach and the existing results via state-space approach is clarified.