  We present in detail a formulation of the shell model as a path integral and
Monte Carlo techniques for its evaluation. The formulation, which linearizes
the two-body interaction by an auxiliary field, is quite general, both in the
form of the effective `one-body' Hamiltonian and in the choice of ensemble. In
particular, we derive formulas for the use of general (beyond monopole) pairing
operators, as well as a novel extraction of the canonical (fixed-particle
number) ensemble via an activity expansion. We discuss the advantages and
disadvantages of the various formulations and ensembles and give several
illustrative examples. We also discuss and illustrate calculation of the
imaginary-time response function and the extraction, by maximum entropy
methods, of the corresponding strength function. Finally, we discuss the
"sign-problem" generic to fermion Monte Carlo calculations, and prove that a
wide class of interactions are free of this limitation.
