Suspensions of colloidal particles with hard-core repulsions and short-ranged attractions are known to exhibit nontrivial behavior. In particular, these systems can form two glasses (a "repulsive glass" and an "attractive glass") at the two extremes of interparticle attractive strength. There are, however, a number of open questions regarding how these processes occur. One involves explaining why modest increases in structural order accompany the pronounced slowing of liquid-state dynamics near structural arrest. Another involves understanding why self-diffusion in glassy fluids occurs much faster than would be predicted from knowledge of the viscosity and the Stokes-Einstein relation. Also, single-particle displacements become heterogeneous near a glass transition, but corresponding structural heterogeneities have been difficult to identify. Collectively, these observations call into question the general prospect of understanding and predicting dynamical behavior of fluids based on structural information. In this talk, we present simulation data of simple model colloidal fluids near both the repulsive and attractive glass transitions. We show that the breakdown of the Stokes-Einstein relationship reflects simple and distinct couplings between structure, viscosity, and diffusivity. We also demonstrate how heterogeneous dynamics correlate with heterogeneous structure.