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A nonperturbative theory means a theory where all the results can be calculated in principle to arbitrary accuracy on a computer. This is really just the same as a well-defined theory, a theory which is mathematically ok, and which makes sense, which isn't incomplete in some way at some high energy or when you do measurements at some high accuracy. An ...

Regarding your specific question: It's (2). Pertubative QFT arises from non-perturbative QFT by Taylor series approximation in the coupling coefficients. Really, non-perturbative QFT should just be called "QFT". But people often try to define a QFT by writing down a perturbative approximation, so we are stuck with this weird terminology.

The standard nonperturbative way (that provided rigorous constructions in 1+1D and 1+2D QFTs) is constructing the Euclidean (imaginary time) field theory as a limit of lattice theories, and then using analytic continuation to real time via the Osterwalder--Schrader theorem. In 1+3D, there is so far no rigorous construction of an interacting QFT, but ...

Instanton calculations involve integrals over collective coordinates. One of these is the instanton size $\rho$. Reliable instanton calculations are those for which the integral over instanton sizes is dominated by small sizes so that (for asymptotically free theories) the coupling constant is small and higher order corrections in the semi-classical ...

Ideally, when we have a differential equation to solve, we will try to solve it analytically. Find explicit functions that encode the variables. The solutions for a harmonic oscillator, for example. The solutions of a potential in Schroedinger's equation. Those are examples of non perturbative solutions. They satisfy the differential ( or integral) ...

Here is my answer from a condensed matter physics point of view: Quantum field theory is a theory that describes the critical point and the neighbor of the critical point of a lattice model. (Lattice models do have a rigorous definition). So to rigorously define/classify quantum field theories is to classify all the possible critical points of lattice ...