TITLE: Which cluster algebras have been categorified? 
QUESTION [17 upvotes]: In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a construction of a cluster category given a quiver with potential whose Jacobian algebra is finite dimensional (in particular, this gives a cluster category for cluster algebras coming from unpunctured surfaces with nonempty boundary). 
My question is simply in what other instances have cluster categories been constructed? In particular, what about cluster algebras with coefficients?  What about other surface cluster algebras?  What about tame cluster algebras?

REPLY [10 votes]: Jan's answer includes many excellent references.  I will try to give a few quick comments.
First of all, although the original Buan-Marsh-Reineke-Reiten-Todorov paper contained some results which were restricted to finite type cluster algebras, in subsequent work of them and others, notably Caldero-Keller, it has been shown that the BMRRT cluster category does indeed categorify any acyclic cluster algebra without coefficients (in particular, including those of tame type).  
Although categorifications are often done in coefficient-free settings, one can also "cheat" by categorifying the cluster algebra in which the coefficients are treated as variables.  For example, this allows one to treat the case of principal coefficients (in some sense, the most important case) for an acyclic cluster algebra using the BMRRT technology.  
I should also point out that the categorifications in question here are what are now sometimes called additive categorifications; Hernandez-Leclerc and Nakajima have a multiplicative categorification which is more in the spirit of what people mean by categorification in other settings: they construct a category with direct sum and tensor product, and show the ring structure induced on the Grothendieck group of their categories is naturally a cluster algebra.