Firstly, this paper reminds the reader of some basic facts about the glassy state, then of the various ways to produce amorphous metals with particular emphasis on the route of vitrification from the melt. Vitrification of an undercooled melt is the most important route from the viewpoint of the application of metallic glasses. We compare diffusion in some metallic glasses with related crystalline metals. Glassy metals, also called metallic glasses, comprise conventional [1] and bulk metallic glasses [2,3]. We remind the reader of the major experimental techniques for diffusion studies in metallic glasses. The paper then reviews our current understanding of diffusion in glassy metals (see also [4,5,6]), including conventional as well as bulk metallic glasses and undercooled melts. We cover the temperature dependence of diffusion in metallic glasses and discuss the spectrum of activation parameters of glassy metals and its difference to the corresponding one of crystalline metals. We mention the pressure dependence and the isotope effect and we discuss tracer diffusion and viscosity diffusion for a bulk metallic glass and its undercooled melt. Finally we mention computer simulations of atomic jump processes. The diffusion mechanism in metallic glasses differs from that in crystalline metals and involves thermally activated, highly collective (chain-like or caterpillar-like) diffusion jumps. Finally, we mention diffusion along shearbands in a plastically deformed glassy metal.