In this paper, new definitions of observability are introduced. They include states and/or inputs estimation, as well as partial or global observability. From these definitions, problems of left invertibility, fault detection, observability, identifiability, unidirectional synchronization,... can be studied in the same context. Moreover, we propose the definitions for both nonlinear continuous-time systems and switched systems. Two examples are introduced to illustrate the efficiency of the proposed definitions and related propositions introduce in the paper.