An introduction to the theory and application of modes -structures that capture the common underlying algebra of convex sets, affine spaces and certain ordered sets. Modes appear in many branches of mathematics, particularly geometry and combinatorics, and have been used in computer science, economics, physics and biology. The initial stage of the theory was set out in the authors' research monograph "Modal Theory" (published in 1985). The present text seeks to provide a more complete theory, the result of research conducted during the subsequent 15 years. It contains a clear introduction to...