We examine the buoyancy-driven motion and deformation of viscous drops through vertical cylindrical capillaries under conditions where interfacial tension effects are important while inertial effects are negligible. Experimental measurements of the terminal velocity of the drops are reported for a range of drop sizes in a variety of two-phase systems, and the drop shapes are quantitatively characterized using digital image analysis. The retarding effect of the capillary wall is found to decrease as the buoyancy force becomes more dominant compared to surface tension, or as the drop fluid becomes less viscous relative to the suspending fluid. However, for a given viscosity ratio, there is a limiting value of the Bond number beyond which the wall effect remains unchanged with further increases in the Bond number. The thickness of the liquid film surrounding large drops is found to be rather insensitive to the value of the viscosity ratio. Theoretical predictions based on the numerical solution of the creeping flow equations using the boundary integral method are shown to be in good agreement with the experimental measurements.