This lecture reviews a series of recent results based on the geometric minimization of the resistance to flow between one point (source, sink) and a volume or an area (an infinity of points). Optimization is achieved by varying the geometric features of the flow path subject to volume constraints. The method is outlined by using the problem of steady volume-point conduction. Optimized first is the smallest elemental volume, which is characterized by volumetric heat generation in a low-conductivity medium, and one-dimensional conduction through a high-conductivity 'channel'. Progressively larger volumes are covered by assemblies of previously optimized constructs. Tree-shaped flow structures spring out of this objective and constraints principle. Analogous problems of fluid flow, and combined heat and fluid flow (convection, trees of fins) are also discussed. The occurrence of similar tree structures in nature may be reasoned based on the same principle (constructal theory) (Bejan, 2000). Copyright (C) 2003 John Wiley Sons, Ltd.