I will review the classical formalism of electronic structure theory that is based on the two-body reduced density matrix and associated representability conditions. The N-representability problem is unsolved and results from computational complexity theory indicate that it won't be solved in any concise form. I hope on the one hand to make clear that it doesn't matter, and on the other hand to outline the challenge of obtaining what could be called anyway a solution to the N-representability problem. Finally I will discuss the computational complexity of the Hartree-Fock problem. As an algebraic problem with general 2-body terms in the hamiltonian (not limited to Coulomb interaction) it is NP-hard, while in the setting of parameterized complexity theory, with the number of electrons as the parameter, it is W[1]-hard.