  We discuss the role of formal deformation theory in quantum field theories
and present various ``higher operations'' which control their deformations,
(generalized) OPEs, and anomalies. Particular attention is paid to
holomorphic-topological theories where we systematically describe and
regularize the Feynman diagrams which compute these higher operations in free
and perturbative scenarios, including examples with defects. We prove
geometrically that the resulting higher operations satisfy expected ``quadratic
axioms,'' which can be interpreted physically as a form of Wess-Zumino
consistency condition for BRST symmetry. We discuss a higher-dimensional
analogue of Kontsevich's formality theorem, which proves the absence of
perturbative corrections in TQFTs with two or more topological directions. We
discuss at some length the relation of our work to the theory of factorization
algebras and provide an introduction to the subject for physicists.
