Algebraic algorithms for multiloop calculations The first 15 years. Whatʹs next?

The ideas behind the concept of algebraic (“integration-by-parts”) algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the corresponding Feynman-parametrized integrands into a form that is optimal for numerical integration, with all the poles in D — 4 explicitly extracted.