Theory of convex structures Presented in this monograph is the current state of the art in the theory of convex structures. The notion of convexity covered here is considerably broader than the classic one; specifically it is not restricted to the context of vector spaces. Classical concepts of order convex sets (Birkhoff) and of geodesically convex sets (Menger) are directly inspired by intuition; they go back to the first half of this century. An axiomatic approach started to develop in the early Fifties. The author became attracted to it in the mid Seventies resulting in the present volume in which graphs appear side by side with Banach spaces classical geometry with matroids and ordered sets with metric spaces. A wide variety of results has been included (ranging for instance from the area of partition calculus to that of continuous selection). The tools involved are borrowed from areas ranging from discrete mathematics to infinite dimensional topology. Although addressed primarily to the researcher parts of this monograph can be used as a basis for a well balanced one semester graduate course