Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms.
It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years from a more geometric and functorial perspective.
First, one must propose a set of axioms which describe the general properties of any mathematical object that deserves to be called a "quantum field theory".
These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space.
In practice, one often uses the Wightman reconstruction theorem, which guarantees that the operator-valued distributions and the Hilbert space can be recovered from the collection of correlation functions.
For the Schwinger functions there is a list of conditions — analyticity, permutation symmetry, Euclidean covariance, and reflection positivity — which a set of functions defined on various powers of Euclidean space-time must satisfy in order to be the analytic continuation of the set of correlation functions of a QFT satisfying the Wightman axioms.