The holomorphic sigma-model is a field theory that exists in any complex dimension that describes the moduli space of holomorphic maps from one complex manifold to another. We introduce the general notion of a holomorphic field theory, which is one that is sensitive to the underlying complex structure of the spacetime manifold (and potentially other geometric input data). Throughout, we rigorously study perturbative quantum field theory in a way that combines the Batalin-Vilkovisky formalism and the effective approach to renormalization. In addition to computing the one-loop anomaly of the holomorphic sigma-model, we study the local operators of the theory using factorization algebras and compute the local index. The final part of this thesis investigates the symmetries present in a general holomorphic theories. We characterize the local central extensions of the two fundamental symmetry algebras, which provide higher dimensional generalizations of the Kac-Moody and Virasoro vertex algebras.