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Motivated by conceptual problems in quantum theories of gravity, thegravitational eikonal approach, inspired by its electromagnetic predecessor,has been successfully applied to the transplanckian energy collisions ofelementary particles and strings since the late eighties, and to string-branecollisions in the past decade. After the direct detection of gravitationalwaves from black-hole mergers, most of the attention has shifted towardsadapting these methods to the physics of black-hole encounters. For suchsystems, the eikonal exponentiation provides an amplitude-based approach tocalculate classical gravitational observables, thus complementing moretraditional analytic methods such as the Post-Newtonian expansion, theworldline formalism, or the Effective-One-Body approach. In this review wesummarize the main ideas and techniques behind the gravitational eikonalformalism. We discuss how it can be applied in various different physicalsetups involving particles, strings and branes and then we mainly concentrateon the most recent developments, focusing on massive scalars minimally coupledto gravity, for which we aim at being as self-contained and comprehensive aspossible.

The gravitational eikonal: from particle, string and brane collisions to black-hole encounters