Matrix factorizations and exotic derived categories

The triangulated category of conventional free or projective matrix factorizations over an affine scheme can be simply defined as their homotopy category, but the constructions of the derived categories of the second kind are needed in order to work either with locally free matrix factorizations over a nonaffine scheme, or with coherent (analogues of) matrix factorizations. I will discuss the definitions of the coderived and absolute derived categories of various classes of matrix factorizations, and their interrelations. Various complements to and strengthenings of Orlov's recent theorem connecting matrix factorizations over a singular nonaffine scheme with the triangulated category of singularities of the zero locus of the superpotential will be given.