Water absorption and wetting at metal surface have received considerable attention due to the important role in many relevant areas including catalysis and corrosion. The glassy surface has unique physical and chemical properties, displaying promising applications in surface science and technology. However, the water wetting of metallic glass surface is less studied than that of crystal metal surface. In this paper, the wetting kinetics of water droplets at the surface of Cu50Zr50 glass is studied by using molecular dynamics simulations. The water droplets show a complete wetting behavior at the glassy surface as in the cases of the CuZr (110) and (110) crystal surfaces. However, the spreading rate of water droplets on the glassy surface is remarkably fast. Despite different spreading rates, the time dependence of the spreading radius for crystal and glass surfaces consistently follows a power law, Rn t with the same exponent n = 7, which conforms with the universal law of the water spreading at non-reactive solid surfaces. An advancing adsorption monolayer of water is formed at the glassy surface, whereas the front of spreading water droplets displays a foot-like morphology at each of the (110) and (110) surfaces. The spreading of water droplets can be described as the process that water molecules diffuse from the droplet surface to the front of the adsorption layer. To reveal the microscopic mechanism of the fast spreading at the glassy surface, the interactions between surface and water are analyzed. We find that the water molecules in the adsorption layer at the glassy surface display a disordered arrangement in contrast to those of the ordered and double-layer structure. The structure of adsorption layer is closely related to the orientations of water molecules in it. The water molecules in the adsorption layer at the glassy surface are mostly parallel to the surface, and those at the crystal surface tend to point to the interiors of droplets. The molecular orientation is proved to determine the relatively weak hydrogen-bond interactions between the adsorption layer and the droplet interior at the Cu50Zr50 glassy surface, thus facilitating the diffusion of water molecules from the droplet surface to the front of the adsorption layer and improving the spreading. On the contrary, the strong interactions associated with the crystal surfaces hinder the droplet from spreading by slowing down the molecular diffusion. The present work provides an insight into the microscopic mechanism of water spreading at metallic glassy surfaces and conduces to in depth understanding the physical and chemical processes associated with metallic-glass/water interfaces.