Recent progress in generation of optical pulses of durations comparable to one optical cycle has presented great opportunities for studies of the fundamental processes in matter as well as time-resolved spectroscopy of ultrafast processes in nonlinear media. It opened up a new area of research in modern ultrafast nonlinear optics and led to appearance of the attosecond science. In parallel, a new research area related to emission from resonant media excited by superluminally propagating ultrashort bursts of electromagnetic radiation has been actively developed over the last few years. In this paper, we review our recent results on theoretical analysis of the Cherenkov-type radiation of a resonant medium excited by few-cycle optical pulses propagating at superluminal velocity. This situation can be realized when an electromagnetic pulse with a plane wavefront incidents on a straight string of resonant atoms or a spot of light rotates at very large angular frequency and excites a distant circular string of resonant dipoles. Theoretical analysis revealed some unusual and remarkable features of the Cherenkov radiation generated in this case. This radiation arises in a transient regime which leads to the occurrence of new frequencies in the radiation spectrum. Analysis of the characteristics of this radiation can be used for the study of the resonant structure properties. In addition, a nonlinear resonant medium excited at superluminal velocity can emit unipolar optical pulses, which can be important in ultrafast control of wave-packet dynamics of matter. Specifics of the few-cycle pulse-driven optical response of a resonant medium composed of linear and nonlinear oscillators is discussed.