Hyperbolization procedures are constructions that turn a simplicial complex into a metric space of non-positive curvature. They were first introduced by Gromov, and later refined by Charney and Davis to produce spaces of strictly negative curvature. In the first part of the talk I will describe some notions of curvature, some hyperbolization procedures, and then showcase some applications in the theory of manifolds. In the second part of the talk I will present joint work with J. Lafont, in which we show that, while the spaces obtained via hyperbolization are often topologically exotic, their fundamental groups are as nice as possible. Namely, these groups admit nice actions on cubical complexes, and are therefore linear over the integers. As an application, we obtain new examples of hyperbolic groups that algebraically fiber.