Most materials attain a glassy state at low temperatures under suitable methods of preparation. This state exhibits the mechanical properties of a solid, but shows microscopic structural disorder(1,2). A comprehensive understanding of the glassy state is, however, still lacking(3). A widespread assumption is that the non-exponential relaxation processes observed in the dynamics of glasses-and also in protein dynamics, protein folding and population dynamics-are (in common with other manifestations of complex dynamics) strongly influenced by the underlying energy landscape associated with the structural configurations that the system may adopt. But concrete evidence for this in studies of glass formation has been scarce. Here we present such evidence, obtained from computer simulations of a model glass-forming liquid. We demonstrate that the onset of non-exponential relaxation corresponds to a well defined temperature below which the depth of the potential-energy minima explored by the liquid increases with decreasing temperature, and above which it does not. At lower temperatures, we observe a sharp transition when the liquid gets trapped in the deepest accessible energy basin. This transition temperature depends on the cooling rate, in a manner analogous to the experimental glass transition. We also present evidence that the barrier heights separating potential-energy minima sampled by the liquid increase abruptly at a temperature above the glass transition but well below the onset of non-exponential relaxation, This identification of a relationship between static, topographic features of the energy landscape and complex dynamics holds the promise of a clearer, possibly thermodynamic, understanding of the glass transition.