The microcirculation in most tissues consists of an intricate network of very narrow tubes. In analyses of blood flow through the microcirculation, inertial effects can be neglected, but continuum models for blood cannot be assumed, since blood is a concentrated suspension of cells with dimensions comparable to vessel diameters. These cells strongly influence blood flow. About 45% of blood volume consists of red blood cells, whose key mechanical properties are known. A red cell has a fluid interior, surrounded by a flexible membrane, which strongly resists area changes, but bends and shears easily. White blood cells are comparable in size but much less numerous. They are less flexible than red cells and capable of active locomotion. Other suspended elements are much smaller than red cells: This review focuses on the mechanics of red cell motion in the microcirculation. Experimental and theoretical studies of blood flow in uniform tubes, bifurcations and networks are discussed. Comparisons between predicted and observed flows in networks imply that resistance to blood flow in living microvessels is higher than that in uniform tubes with corresponding diameters. Living microvessels have non-uniform geometries, and red cells must deform continually to traverse them. Theoretical results are presented implying that these transient deformations contribute to increased flow resistance in the microcirculation.