  In this thesis, we explore two approaches to string phenomenology. In the
first half of the work, we investigate M-theory compactifications on spaces
with co-dimension four, orbifold singularities. We construct M-theory on
C^2/Z_N by coupling 11-dimensional supergravity to a seven-dimensional
Yang-Mills theory located on the orbifold fixed-plane. The resulting action is
supersymmetric to leading non-trivial order in the 11-dim Newton constant. We
thereby reduce M-theory on a G2 orbifold with C^2/Z_N singularities, explicitly
incorporating the additional gauge fields at the singularities. We derive the
Kahler potential, gauge-kinetic function and superpotential for the resulting
N=1 four-dimensional theory. Blowing-up of the orbifold is described by a Higgs
effect and the results are consistent with the corresponding ones obtained for
smooth G2 spaces. Further, we consider flux and Wilson lines on singular loci
of the G2 space, and discuss the relation to N=4 SYM theory.
  In the second half, we develop an algorithmic framework for E8 x E8 heterotic
compactifications with monad bundles. We begin by considering cyclic Calabi-Yau
manifolds where we classify positive monad bundles, prove stability, and
compute the complete particle spectrum for all bundles. Next, we generalize the
construction to bundles on complete intersection Calabi-Yau manifolds. We show
that the class of positive monad bundles, subject to the heterotic anomaly
condition, is finite (~7000 models). We compute the particle spectrum for these
models and develop new techniques for computing the cohomology of line bundles.
There are no anti-generations of particles and the spectrum is manifestly
moduli-dependent. We further study the slope-stability of positive monad
bundles and develop a new method for proving stability of SU(n) vector bundles.
