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These are the extended lecture notes for a minicourse presented at the IS\~ao Paulo School on Gravitational Physics discussing theBondi--Metzner--Sachs (BMS) group, the group of symmetries at null infinity onasymptotically flat spacetimes. The BMS group has found many applications inclassical gravity, quantum field theory in flat and curved spacetimes, andquantum gravity. These notes build the BMS group from its most basicprerequisites (such as group theory, symmetries in differential geometry, andasymptotic flatness) up to modern developments. These include its connectionsto the Weinberg soft graviton theorem, the memory effect, its use to constructHadamard states in quantum field theory in curved spacetimes, and other ideas.Advanced sections briefly discuss the main concepts behind the infraredtriangle in electrodynamics, superrotations, and theDappiaggi--Moretti--Pinamonti group in expanding universes with cosmologicalhorizons. New contributions by the author concerning asymptotic (conformal)Killing horizons are discussed at the end.

Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics