In this paper, the problems of state estimation are investigated for switched linear systems with average dwell time (ADT) switching in both continuous-time and discrete-time contexts. First, a set of mode-dependent Luenberger-type observers is designed subject to the ADT switching that is synchronous with the switching of the estimated systems. Then, a more practical case of the delayed observers is also considered, which implies that the switching of the multiple-mode observer to be designed has a lag to the switching of the estimated systems. In this case, the asynchronous switching signals are combined as a preliminary attempt, upon which sufficient conditions for the existence of the Luenberger-type observers and the corresponding ADT switching are formulated in terms of a set of linear matrix inequalities. Finally, the proposed approaches are applied to the state estimation of electronic circuits to demonstrate the effectiveness and applicability.