A brief general survey of the current state of the art of quantum chemistry is given with some aspects also towards the future. It is emphasized that, if one wants to incorporate such concepts as temperature, entropy, and free energy into quantum chemistry, it is necessary to make a transition from pure quantum mechanics based on wavefunctions to the more general “quantum statistics” based on density matrices and system operators. In addition to the Schrödinger equations, one obtains the Liouville equations, and it is shown that both the time-dependent and the time-independent equations may be solved in both cases by using analogous Hilbert-space methods. Some of the methods for solving the time-independent eigenvalue problems are reviewed, and the need for giant “number crunchers” in this connection are discussed. It is shown that the resolvent methods combined with the “inner projection” technique for the calculations provide a powerful tool for handling the eigenvalue problems in the future in both the Hamiltonian and Liouvillian formalisms. It is stressed that, by going over to supercomputers, one may gain a factor of 100, and that one may gain another factor of 100 by going over to more powerful theoretical methods; however, for programming reasons, it will take a long time before one can reach the combined efficiency factor 100 ✗ 100.