On the basis of a new theory of finite-temperature magnetism in amorphous materials and liquids, it is shown that amorphous iron forms an itinerant-electron spin glass at low temperatures because of the nonlinear magnetic couplings between iron local moments and local-environment effects on the amplitude of the local moments. The results explain well the recent experimental data of spin glasses in amorphous ${\mathrm{Fe}}_{93}$${\mathrm{La}}_{7}$ and ${\mathrm{Fe}}_{92}$${\mathrm{Zr}}_{8}$ alloys.