A brief but clear and complete account is given for two analytical methods of calculating correlation functions in quantum statistical physics from first principles--the widely used method of two-time temperature Green's functions (the GF method) and a new, ``direct algebraic'' (DA) method. The mathematical and technical clarity and simplicity of the DA method and its resulting practical value are demonstrated for the five most widely used models in quantum statistical physics. Since the DA method is an exactly self-consistent method (in the sense that the expansion coefficients in the equations of motion are chosen from the requirement that the Jacobi operator identity be satisfied exactly), it in principle affords the possibility of an internal check, which is not possible in the GF method. Like the GF method, the DA method permits calculation of the spectra of possible elementary excitations and, hence, of the density of single-particle energy states corresponding to them.