The physically-inspired theory of conformal blocks allows one to construct vector bundles on moduli spaces of curves with remarkable geometric and combinatorial properties. This theory uses as input the representations of some non-commutative algebras. A classical example is provided by the representations of affine Lie algebras, and the resulting vector bundles have been studied at great length in the last thirty years, yielding extraordinary insights on moduli spaces of curves. In this talk, I will present how some fundamental results of the classical theory of conformal blocks extend to the more general setting provided by replacing affine Lie algebras with vertex operator algebras. Specifically, I will discuss an extended factorization property and new cohomological field theories. This is joint work with Chiara Damiolini and Angela Gibney.