Whether in a perturbative or fully nonlinear context, the formulation of dynamics in gravitation relies on a suitable notion of an initial value problem. These come in various flavors, each with relative advantages and disadvantages depending on the particular application at hand. Among these, an attractive possibility is the hyperboloidal initial value problem, in which spacetime is foliated by leaves that terminate at null infinity. Such hyperboloidal foliations are ideally suited to treat outgoing radiation, and are thus finding widespread use in the modern era of gravitational wave astronomy. In this presentation we will give an accessible introduction to all of this, with the aim of touching upon various aspects to be covered in detail during the workshop.