What are the noncommutative rings that are most analogous to polynomial rings?  One class of such rings are the regular algebras first defined by Artin and Schelter in 1987.  Since then such algebras have been extensively studied.  I will give a survey of these interesting examples and some results about them.  I will also mention recent work of Manny Reyes and myself on the more general class of twisted Calabi-Yau algebras.