Theoretical models involving acoustic solitons in elastic media with dispersion were first developed for nonlinear longitudinal waves in rods [Ostrovsky and Sutin (1975); Samsonov (1984)]. Recently, the interest to this problem has considerably grown. At the same time, experimental realizations of elastic solitons are scarce [e.g., the works by Lazaridi and Nesterenko (1985) where strong (nonacoustic) solitons were created in contacting spheres, and by Dreiden et al. (1989), where a localized elastic pulse was observed in a rod]. Here, possibilities of soliton existence in different elastic media and some practically interesting values of their parameters are discussed. Three basic models are considered: (1) a thin elastic rod in which dispersion is due to the finiteness of its diameter; (2) a porous medium where dispersion is due to resonance properties of pores (similar to the known mechanism in a liquid with bubbles); and (3) a grainy medium with random packing of spherical grains in which nonlinearity is due to grain contacts and dispersion is due to finite grain sizes. It is shown that in all these cases, acoustic solitons may exist under realistic conditions. Possible realizations of such waves and their applications for medium--structure diagnostics are also evaluated.