Hall algebras play an important role in representation theory, algebraic geometry and combinatorics. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. We consider twisted and extended Hall algebras of triangulated categories and note that in some cases they are well-defined even when their non-extended counterparts are not. We show that each exact category with weak equivalences with an appropriate extra structure naturally gives rise a twisted extended Hall algebra of its homotopy category. If time permits, we will discuss the relation of this construction to graded quiver varieties and to categorification of modified quantum groups.