Glass formation in colloidal suspensions has many of the hallmarks of glass formation in molecular materials. For hard-sphere colloids, which interact only as a result of excluded volume, phase behaviour is controlled by volume fraction, phi; an increase in phi drives the system towards its glassy state, analogously to a decrease in temperature, T, in molecular systems. When phi increases above phi* approximately 0.53, the viscosity starts to increase significantly, and the system eventually moves out of equilibrium at the glass transition, phi(g) approximately 0.58, where particle crowding greatly restricts structural relaxation. The large particle size makes it possible to study both structure and dynamics with light scattering and imaging; colloidal suspensions have therefore provided considerable insight into the glass transition. However, hard-sphere colloidal suspensions do not exhibit the same diversity of behaviour as molecular glasses. This is highlighted by the wide variation in behaviour observed for the viscosity or structural relaxation time, tau(alpha), when the glassy state is approached in supercooled molecular liquids. This variation is characterized by the unifying concept of fragility, which has spurred the search for a 'universal' description of dynamic arrest in glass-forming liquids. For 'fragile' liquids, tau(alpha) is highly sensitive to changes in T, whereas non-fragile, or 'strong', liquids show a much lower T sensitivity. In contrast, hard-sphere colloidal suspensions are restricted to fragile behaviour, as determined by their phi dependence, ultimately limiting their utility in the study of the glass transition. Here we show that deformable colloidal particles, when studied through their concentration dependence at fixed temperature, do exhibit the same variation in fragility as that observed in the T dependence of molecular liquids at fixed volume. Their fragility is dictated by elastic properties on the scale of individual colloidal particles. Furthermore, we find an equivalent effect in molecular systems, where elasticity directly reflects fragility. Colloidal suspensions may thus provide new insight into glass formation in molecular systems.