  The target space geometry of abelian vector multiplets in ${\cal N}= 2$
theories in four and five space-time dimensions is called special geometry. It
can be elegantly formulated in terms of Hessian geometry. In this review, we
introduce Hessian geometry, focussing on aspects that are relevant for the
special geometries of four- and five-dimensional vector multiplets. We
formulate ${\cal N}= 2$ theories in terms of Hessian structures and give
various concrete applications of Hessian geometry, ranging from static BPS
black holes in four and five space-time dimensions to topological string
theory, emphasizing the role of the Hesse potential. We also discuss the r-map
and c-map which relate the special geometries of vector multiplets to each
other and to hypermultiplet geometries. By including time-like dimensional
reductions, we obtain theories in Euclidean signature, where the scalar target
spaces carry para-complex versions of special geometry.
