This lecture will start with a brief survey of ancient ideas about blood flow, culminating in the West with William Harvey’s convincing demonstration – before the invention of the microscope – that the blood circulates in the body. The mechanics of that circulation will be described, from the high-pressure arteries to the low-pressure veins. Highlights will be: the propagation of the pressure pulse in arteries, the disturbance to smooth flow caused by the complex geometry of arteries (and its probable influence on arterial disease), the fact that blood cells have to be deformed and travel in single file in the smallest capillaries, and the interesting effects of gravity on the venous return to the heart in upright animals, notably those with long necks and legs – giraffes and dinosaurs. Tim Pedley is an applied mathematician whose research has been devoted to Biological Fluid Dynamics, both internal (e.g. blood flow) and external (e.g. micro-organism swimming), for over 50 years . He is Emeritus G I Taylor Professor of Fluid Mechanics at Cambridge, and has served as Chairman of the World Council for Biomechanics, President of the International Union of Theoretical and Applied Mechanics and President of the Institute of Mathematics and its Applications.