Classical invariant theory studies the ring of invariants $\Bbbk[x_1, \dots, x_n]^G$ under the action of a group $G$ on a commutative polynomial ring ${\Bbbk}[x_1, \dots, x_n]$. To extend this theory to a noncommutative context, we replace the polynomial ring with an Artin-Schelter regular algebra $A$ (that when commutative is isomorphic to a commutative polynomial ring), and study the invariants $A^G$ under the action of a finite group, or, more generally, a finite dimensional Hopf algebra. We will discuss some open questions on generalizing classical results to this context.