The aim now is to answer what really defines a quantum mechanical system or a quantum field theory, which in a sense is "field degrees of freedom" (quotes because this is very non-trivial physically) that we obtain from some classical mechanical system or a classical field theory. A classical theory consists of a phase space $T^{*}M$ that is a symplectic manifold $(\mathbb{R}^{2n}, \omega )$ and observables are functions $f\in C^{\infty }(T^{*}M)$ that are elements of a Poisson algebra $\mathcal{P}$. which comes with Poisson brackets $\{\cdot , \cdot \} : C^{\infty }\times C^{\infty }\to C^{\infty }(M)$. Quantum observables are then going to be operators on some $\mathcal{A}(\mathcal{H})$ with a commutator $[\cdot, \cdot ]$ on $\mathcal{H}$. Quantization is then just a function $Q: C^{\infty }(\mathbb{R}^{2n})\to L(\mathcal{H})$, space of linear operators on $\mathcal{H}$. In stronger operator algebraic terms we would actually be more careful about what we do with this (i.e. selecting $\mathcal{B}(\mathcal{H})$), but for now we can be more superficial. So the problem is really just going from this classical phase space to a quantum Hilbert space. Note that I specifically said "quantum Hilbert space", since it is usually easier to obtain a pre-quantum Hilbert space from pre-quantization, where we identify from our Poisson manifold a collection of a hermitian line bundle, a connection on this line bundle and a Hilbert space that is etymologically one but not the actual constrained Hilbert space -- in this abuse of terminology one can consider this pre-quantum Hilbert space the auxiliary Hilbert space, on which polarization is done. I will not get into what polarization actually does for now, but this is typically the actual quantization problem. After polarization, we will end up with the actual quantum Hilbert space, which would be the quantised theory.
So what we did here is called geometric quantization, which is only one of many sorts of quantization. The aim here is of course to end up with a Hilbert space. In a similar fashion, you could ask what happens with field theories. A classical field theory, after all, is also equipped with Poisson algebras and eventually we want to quantize this. Since QFTs describe algebras of observables, you could define it as really a functor from categories of globally hyperbolic manifolds to those of $C^{*}$-algebras satisfying the Haag-Kastler axioms. However, the obvious issue is that field theories have infinite degrees of freedom and therefore the phase space is infinite dimensional. In such cases you would do the so-called refined algebraic quantization, which will be a different post eventually. This was in fact, touched upon in the Hilbert space of de Sitter quantum gravity paper by Suvrat et al, without realising the full technicalities of what they were doing! On the other hand, a more suitable general scheme is that of deformation quantization where the Poisson brackets are "deformed" into commutators. Then there is stochastic quantization and more interestingly, Berezin-Toeplitz quantization which has to do with compact symplectic manifolds and Kostant–Souriau operators, which are related to the Hamiltonian vector field in a certain condition, which I am not entirely sure about.