A Calabi-Yau algebra is a noncommutative analogue of the coordinate ring of a Calabi-Yau variety. It is well-known that if $G$ is a group acting on a Calabi-Yau algebra $A$, then the smash product $A \#G$ remains Calabi-Yau under sufficiently good conditions. However, there are cases in which a smash product $A \# G$ may become Calabi-Yau even if $A$ is not Calabi-Yau. We will explain how this can occur by studying the more general notion of a \emph{skew Calabi-Yau algebra}. This is joint work with D.~Rogalski and J.J.~Zhang.