I think the best way to put my question is the following: what (fermionic) theories make use of Grassmann variables?
Let me clarify my question a little further. I remember some discussions in quantum mechanics books about many-particle systems and, in particular, many-particle fermions. At least in the cases I know, these discussions are commonly superficial in the sense that they do not intend to solve anything, but rather they aim to introduce the mathematical settings of such systems, i.e. how to construct the Fock space, how to define the Hamiltonian etc. The discussion often ends there and usually no Grassmann algebra is used.
The first time I saw the use of Grassmann variables in a physics model was in QFT, where Dirac fields are quantized by using these variables.
However, the Grassmann algebra is a whole universe and it is certainly not just applicable in the Dirac field quantization. In fact, many references like this one make use of Grassmann variables alone to define a theory with a Hamiltonian, creation and operators and so on. This is a fermionic theory as the name suggests, but I cannot understand what it describes, that is, what physical systems are described by this formalism.
So, back to my question. What are these fermionic models described by Grassmann variables? Are they necessarily field theories? (Why is not used in quantum mechanics of many-particles systems otherwise?)
Remark: I obviously do not expect an answer which lists every possible fermionic theory which is described by Grassmann variables. Instead, I'd really like to know how one passes from a general physical theory to its description via Grassmann algebras. If possible, what are these Grassmann integrals trying to calculate.