Properties and possible applications of ultra-short (femto- to pico-second) electromagnetic mono- and sub-cycle waves are studied. First, the structural characteristics of these waves are examined. Transient pulses of circularity polarized waves have more complicated structures than those of plane polarized ones. In addition, a pair of colliding pulses may locally and instantaneously generate purely electric or magnetic fields. A general method is presented to describe linear characteristics of these waves propagating through same homogeneous, dispersive media such as air, plasma and optical fibers. Although their spectra are ultra-wideband, should their time duration be short enough, these pulses can in some cases propagate through dispersive media for relatively long distances, approximately maintaining their original waveforms. In particular, such pulses traveling in a plasma shows a characteristic quite opposite to that of ordinary wave packets.