  We review the spectral cover formalism for constructing both U(n) and SU(n)
holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which
admit a section. We discuss the allowed bases of these three-folds and show
that physical constraints eliminate Enriques surfaces from consideration.
Relevant properties of the remaining del Pezzo and Hirzebruch surfaces are
presented. Restricting the structure group to SU(n), we derive, in detail, a
set of rules for the construction of three-family particle physics theories
with phenomenologically relevant gauge groups. We show that anomaly
cancellation generically requires the existence of non-perturbative vacua
containing five-branes. We illustrate these ideas by constructing four explicit
three-family non-perturbative vacua.
