Time-delay systems form a class of infinite-dimensional systems, inducing severe challenges from a control perspective. In the chapter, we first review basic properties ofFeedback time-delay systems and shed light on these limitations by means of the design of staticStatic outputFeedback feedbackFeedback controllers. For input-delay systems, we then make a survey of two complementary control techniques that allow to render the closed-loop system in some sense finite dimensional. The first one is based on including predictors to compensate delays in the control loop. The second, much less known control-induced reduction technique, consists of using periodicPeriodic output feedback. We show that an appropriate design of the time-varying gains renders the monodromy operator of the closed-loop system finite dimensional. Moreover, this matrix can be made nilpotent or even zero, leading to fixed-time stability. We conclude the chapter with a discussion on the related problem of observation in a guaranteed, prescribed time.