The paper focuses on a long standing problem which consists in identifying those time-delay systems which can be transformed into a delay-free system by a suitable change of state coordinates. Both linear and nonlinear systems are considered. It is shown that the so-called cyclic vectors introduced by Olbrot and co-workers are rehabilitated through the actual control inputs or through some virtual inputs which render the system fully controllable or accessible. Whereas the current literature includes solutions for linear systems, no result is available for nonlinear time-delay systems yet.