  Motivated by conceptual problems in quantum theories of gravity, the
gravitational eikonal approach, inspired by its electromagnetic predecessor,
has been successfully applied to the transplanckian energy collisions of
elementary particles and strings since the late eighties, and to string-brane
collisions in the past decade. After the direct detection of gravitational
waves from black-hole mergers, most of the attention has shifted towards
adapting these methods to the physics of black-hole encounters. For such
systems, the eikonal exponentiation provides an amplitude-based approach to
calculate classical gravitational observables, thus complementing more
traditional analytic methods such as the Post-Newtonian expansion, the
worldline formalism, or the Effective-One-Body approach. In this review we
summarize the main ideas and techniques behind the gravitational eikonal
formalism. We discuss how it can be applied in various different physical
setups involving particles, strings and branes and then we mainly concentrate
on the most recent developments, focusing on massive scalars minimally coupled
to gravity, for which we aim at being as self-contained and comprehensive as
possible.
