  Artin-Schelter regular algebras can be thought of as noncommutative versions
of commutative polynomial rings, modeled after the special homological
properties polynomial rings have as graded rings. First defined by Artin and
Schelter in 1987, their introduction formed the beginning of the subject of
noncommutative projective geometry. Artin-Schelter regular algebras have
continued to play a large role in that subject, since geometrically they
represent noncommutative (weighted) projective spaces.
  This is a survey of Artin-Schelter regular algebras, based on a talk at the
2022 meeting "Recent Advances and New Directions in the Interplay of
Noncommutative Algebra and Geometry" at the University of Washington, in honor
of the 65th birthday of S. Paul Smith. We review the earliest foundational
results in the subject, and then describe some of the major themes of the last
30 years of research.
