Coherent sources of the short–wavelength (2 - 120 nm) range are increasingly being used both in fundamental research and for practical purposes. One of the most effective methods of generating such extreme ultraviolet radiation is the generation of high-order harmonics of ultrashort laser pulses. This review presents studies of the generation of high-order harmonics of femtosecond pulses in laser-induced plasma, starting with the first demonstration of this process under optimal conditions of plasma formation (Optics Letters 30, 768-770, 2005) and up to the most recent studies. The achievements of this method of converting the wavelength of intense laser sources into the extreme ultraviolet range of the spectrum are analyzed. Special attention is paid to new techniques to increase the efficiency of harmonic conversion. In particular, successes are demonstrated in such areas as (a) resonant amplification of single harmonics, (b) amplification of harmonic groups caused by quasi-phase matching of interacting waves in various spectral ranges, (c) two- and three-color plasma pumping to generate odd and even harmonics and parametric waves, (d) the use of quantum dots and nanoparticles for amplification of harmonics of lower orders and other results. The prospects and potential applications of this method of generating coherent shortwave radiation in the extreme ultraviolet region of the spectrum are discussed.