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In 1968, Veselago posed that media with simultaneous negative values of the permittivity and permeability would exhibit a negative index of refraction (Soviet Physics Uspekhi, vol. 10, pp. 509-514, 1968.). Due to the left-handed relationship between the electric field, magnetic field, and wave vector in these materials he termed them left-handed media (LHM). These LHM are expected to have many unusual properties such as inverted Snell's law, Doppler shift and Cherenkov radiation. However, until recently, such materials were unavailable and these predictions remained uncorroborated for nearly thirty years. As with any new field, the topic of the LHM suffers from some misconceptions and misunderstandings. One such misconception has to do with the notion of negative group velocity which has been closely linked with these media (R.A. Shelby et al, Appl. Phys. Lett., vol. 78, pp. 489-91, 2001) and in fact has been used as equivalent to the term LHM. In this paper, the existence and meaning of negative group velocity in LHM are discussed. It is shown that similar to normal right-handed media (RHM), LHM can also exhibit a negative group velocity and group delay within the region of anomalous dispersion (/spl delta/n//spl delta//spl omega/<0). On the other hand, in the experiments presented so far, the researchers have only measured positive group velocity and negative phase velocity corresponding to so called backward waves (BWs).