The study of supercooled liquids and glasses remains one of the most divisive and divided fields in modern physics. Despite a vast amount of effort and research time invested in this topic, the answers to many central questions remain disputed and incomplete. However, the link between the behaviour of supercooled liquids and their energy landscapes is well established and widely accepted. Understanding this link would be a key step towards resolving many of the mysteries and controversies surrounding the glass transition. Therefore the study of glassy energy landscapes is an important area of research. In this thesis, I report some of the most detailed computational studies of glassy potential energy landscapes ever performed. Using geometry optimisation techniques, I have sampled the local minima and saddle points of the landscapes for several supercooled liquids to analyse their dynamics and thermodynamics. Some of my analysis follows previous work on the binary Lennard-Jones fluid (BLJ), a model atomic liquid. BLJ is a fragile glass former, meaning that its transport coefficients have super-Arrhenius temperature dependence, rather than the more usual Arrhenius behaviour exhibited by strong liquids. The difference in behaviour between these two classes of liquid has previously been attributed to differing degrees of structure in the relevant energy landscapes. I have studied models for both fragile and strong glass formers: the molecular liquid ortho-terphenyl (OTP) and viscous silica (SiO2) respectively. My results for OTP agree closely with trends observed for BLJ, suggesting that the same diffusion mechanism is applicable to fragile molecular liquids as well as to atomic. However, the dynamics and energy landscape of OTP are made complicated by the molecular orientational degrees of freedom, making the analysis more challenging for this system. Dynamics of BLJ, OTP and silica are all dominated by cage-breaking events: structural rearrangements in which atoms change their nearest neighbours. I propose a robust and general method to identify cage breaks for small rigid molecules, and compare some properties of cage breaks between strong and fragile systems. The energy landscapes of BLJ and OTP both display hierarchical ordering of potential energy minima into metabasins. These metabasins can be detected by the cage-breaking method. It has previously been suggested that metabasins are responsible for super-Arrhenius behaviour, and are absent from the landscapes of strong liquids such as SiO2. My results indicate that metabasins are present on the silica landscape, but that they each contain fewer minima than metabasins in BLJ or OTP. Metabasins are associated with anticorrelated particle motion, mediated by reversed transitions between minima of the potential energy landscape. I show that accounting for time-correlation of particle displacement vectors is essential to describe super-Arrhenius behaviour in BLJ and OTP, but also required to reproduce strong behaviour in silica. I hypothesise that the difference between strong and fragile liquids arises from a longer correlation timescale in the latter case, and I suggest a number of ways in which this proposition could be tested. I have investigated the effect on the landscape of freezing the positions of some particles in a BLJ fluid. This “pinning” procedure induces a dynamical crossover that has been described as an equilibrium “pinning transition”, related to the hypothetical ideal glass transition. I show that the pinning transition is related to (and probably caused by) a dramatic change in the potential energy landscape. Pinning a large fraction of the particles in a supercooled liquid causes its energy landscape to acquire global structure and hence structure-seeking behaviour, very different from the landscape of a typical supercooled liquid. I provide a detailed description of this change in structure, and investigate the mechanism underlying it. I introduce a new algorithm for identifying hierarchical organisation of a landsape, which uses concepts related to the pinning transition but is applicable to unpinned liquids as well. This definition is complementary to metabasins, but the two methods often identify the same higher-order structures. The new “packings” algorithm offers a route to test thermodynamic theories of the glass transition in the context of the potential energy landscape. Over the course of this thesis, I discuss several different terms and methods to identify higher-order structures in the landscapes of model glass formers, and investigate how this organisation varies between different systems. Although little variation is immediately apparent between most glassy landscapes, deeper analysis reveals a surprising diversity, which has important implications for dynamical behaviour in the vicinity of the glass transition.