2-dimensional quantum field theory by a symmetry group admits an elegant description in terms of defect lines and their junction fields. This perspective offers a natural generalization of the concept of an orbifold, in which the role of the symmetry group is replaced by a defect with the structure of a (symmetric) separable Frobenius algebra. In this talk I will focus on the case of Landau-Ginzburg models, in which defects are described by matrix factorizations. After introducing the generalized twisted sectors and discussing topological bulk and boundary correlators in these sectors, I will present a simple proof of the Cardy condition and discuss some further consistency checks on the generalized orbifold theory. This talk is based on arXiv:1307.3141 with Ilka Brunner and Nils Carqueville.