Non-commutative cluster algebras of generalized surface type

I will explain work in progress on a new class of non-commutative cluster algebras which generalize the work of Berenstein and Retakh on non-commutative surfaces. Time permitting, I will briefly discuss the connection with A-type cluster coordinates (Fock-Goncharov coordinates) for decorated representations into Spin(p,q) and the relation to Theta-positivity.