Colloidal suspensions are ubiquitous in industrial and technological applications; moreover, the precise control over the interparticle interactions allows such suspensions to serve as excellent model systems for a variety of complex fluids and soft materials. In addition, the large size of colloidal particles enables direct visualization of local structure. In this colloquium I will discuss two studies relating the structure of non-equilibrium colloidal suspensions to their flow properties. First, we use confocal microscopy to investigate relationships between structure and dynamics near the hard-sphere colloidal glass transition, one of the fundamental unsolved problems in condensed matter physics. By identifying slowly-relaxing regions within our samples, we show that the structure of these clusters is correlated to the macroscopic mechanical properties of the suspension. Second, we use microscopy, light scattering, and rheology to characterize the properties of dense colloidal gels formed via arrested phase separation. These dense gels exhibit structural properties of both fractal colloidal gels and colloidal glasses, yet their mechanical properties are strikingly different from either. Collectively, these measurements yield fundamental insight into the relationship between structure and dynamics in non-equilibrium colloidal suspensions.