The geometry of Hessian structures is a fascinating emerging field of research. It is in particular a very close relative of Kählerian geometry, and connected with many important pure mathematical branches such as affine differential geometry, homogeneous spaces and cohomology. The theory also finds deep relation to information geometry in applied mathematics. This systematic introduction to the subject first develops the fundamentals of Hessian structures on the basis of a certain pair of a flat connection and a Riemannian metric, and then describes these related fields as applications of the theory.Contents:Affine Spaces and ConnectionsHessian StructuresCurvatures for Hessian StructuresRegular Convex ConesHessian Structures and Affine Differential GeometryHessian Structures and Information GeometryCohomology on Flat ManifoldsCompact Hessian ManifoldsSymmetric Spaces with Invariant Hessian StructuresHomogeneous Spaces with Invariant Hessian StructuresHomogeneous Spaces with Invariant Projectively Flat ConnectionsReadership: Mathematicians and mathematics graduate students.