In this paper, infinite dimensional systems described by convolutional equations and, in particular, by delay-differential equations are considered. Different controllability notions for this class of linear time-invariant systems are discussed and compared. A characterization of spectral controllability for a system whose trajectories satisfy a homogeneous system of independent convolutional equations is given. This result extends an analogous result which was known to hold for difference or differential equations. Finally, for a particular class of systems, including systems in the state space form, it is shown that a well-known theorem, which states the equivalence of spectral controllability and the existence of an image representation, holds true. An example is presented showing that this result is false for generic delay-differential systems as soon as there are two noncommensurate delays.