A notion of finite time observer for partially observed deterministic control systems is introduced; it is shown how such observers are obtained by probabilistic-variational methods. Observability plays an essential role. The procedure is carried out for finite-state discrete-time systems and continuous-time nonlinear systems, and the observers obtained are respectively finite and infinite dimensional. A simple algorithm implementing the observer for finite-state systems is described. An observability grammian for nonlinear systems is introduced and is used to study the time evolution of sets of indistinguishable points, as well as local properties of the function satisfying the observer equation. Finally, the results are specialized to bilinear systems.