Originally introduced by Eisenbud in the context of commutative algebra, matrix factorizations have since earned a prominent role in mathematical physics. In this talk I will describe recent work using matrix factorizations to define a cohomological field theory given the input data of a gauged linear sigma model. This construction gives a new description of the virtual class in Gromov-Witten theory, FJRW theory, and so-called hybrid models. This is joint work with Ciocin-Fontanine, Favero, Guere, and Kim