Understanding the slowing down of liquids as they approach the glass transition is an enduring problem in glass science. The early stages of glassy slowdown are particularly difficult to describe theoretically and have led to several and often contradictory interpretations. In this talk, I will present recent results on the so-called onset of glassiness, where liquids first start to display glassy dynamics upon cooling. In particular, I will investigate the relationship between the onset, local structure and thermodynamic metastability, both in computer models and actual experiments. The extent to which current theories of the glass transition, including mode-coupling theory, can account for the features of the onset of glassiness will be critically discussed.