Unlike the well-defined long-range order that characterizes crystalline metals, the atomic arrangements in amorphous alloys remain mysterious at present. Despite intense research activity on metallic glasses and relentless pursuit of their structural description, the details of how the atoms are packed in amorphous metals are generally far less understood than for the case of network-forming glasses. Here we use a combination of state-of-the-art experimental and computational techniques to resolve the atomic-level structure of amorphous alloys. By analysing a range of model binary systems that involve different chemistry and atomic size ratios, we elucidate the different types of short-range order as well as the nature of the medium-range order. Our findings provide a reality check for the atomic structural models proposed over the years, and have implications for understanding the nature, forming ability and properties of metallic glasses.