This paper raises the question of controllability (or more precisely, reachability) of sampled-data systems, i.e., systems composed of a continuous-time plant controlled by a digital controller. The question is formulated from the continuous-time perspective, and the study is carried out in a functional analytic framework. The novelty of the study lies in the infinite-dimensionality of the underlying physical state signals which comes from the fact that the plant actually evolves over a temporal continuum. Different concepts of controllability related directly to the continuous-time behavior of sampled-data systems are introduced, namely exact, approximate, and null controllability. It is shown that the two former notions never occur in sampled-data systems, while the latter notion is a generic property for these systems. Some facts regarding the fundamental input/output structure of sampled-data systems from the control viewpoint are drawn from these results. The spectrum distribution of sampled-data feedback systems is also characterized, and its connection to the controllability properties is pointed out. By way of contrast, the system-theoretic viewpoint adopted in this paper departs from the finite-dimensional state-space system approach which has been taken at the outset in the lifting framework, and in which the pure discrete-time state of the "stroboscopic" model is chosen as the state of the hybrid sampled-data systems. Such a choice for a minimal finite-dimensional state space is justified by the external (input/output) treatment of sampled-data systems. We stress the fact that this paper is situated entirely at the internal level and takes a closer look into the internal continuous-time dynamics of sampled-data systems.