The aim of this paper is to study the observability for systems described by first-order evolution equations and for those described by second-order evolution equations in the case of discrete-time observations. For the systems with a finite number of sensors we present necessary and sufficient conditions for observability. We show that these distributed parameter systems are never finite-step observable. We give the restricted sets of the initial-state spaces whose elements areN-step observable. We also investigate the relations between the systems with discrete-time observations and the systems with continuous-time observations from the viewpoint of observability Moreover, we see the essential difference between the parabolic case and the hyperbolic case.