I will present an introduction to the geometry of hyperbolic polynomials. A hyperbolic polynomial h is a homogeneous polynomial in several real variables that admits a hyperbolic point, i.e. a point where the negative Hessian of h has Minkowski signature. Such polynomials arise in the theory of Kähler cones as the volume on the real (1,1)-cohomology. In the cubic case their study is further motivated by supergravity. The r- and c-map constructions allow one to construct explicit examples of projective special Kähler and quaternionic Kähler manifolds. I will introduce the basic concepts and theory needed for this topic, highlight recents results, and discuss some challenges in ongoing research. At the end of the talk I will discuss a selection of open problems and possible ways to solve them.