This paper provides a review of theoretical and experimental research to date on the constrained steady state solidification of dilute binary alloy matrix composites. Steady state solidification can be experimentally observed in composites which contain a cylindrical reinforcement aligned parallel to the growth direction of samples solidified in a Bridgman furnace. Such experiments have been performed by several authors, using metallic or succinonitrile-based matrices. It is found that plane front, cellular, and dendritic solidification modes, and transitions between solidification modes, are all strongly influenced by the reinforcement if its dimensional scale is below those of the solidifying unreinforced matrix microstructure. General features of these phenomena are reviewed, together with their current theoretical interpretation.