Cluster categories for algebras of global dimension 2 and quivers with potential

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\le 2$. We construct a triangulated category ${\text{Cc}}_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When ${\text{Cc}}_A$ is $\text{Hom}$-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{"o}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category ${\text{Cc}}_{(Q,W)}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra $\text{Jj}(Q,W)$.