  Motivated by the simplicity and direct phenomenological applicability of
field-theoretic orbifold constructions in the context of grand unification, we
set out to survey the immensely rich group-theoretical possibilities open to
this type of model building. In particular, we show how every maximal-rank,
regular subgroup of a simple Lie group can be obtained by orbifolding and
determine under which conditions rank reduction is possible. We investigate how
standard model matter can arise from the higher-dimensional SUSY gauge
multiplet. New model building options arise if, giving up the global orbifold
construction, generic conical singularities and generic gauge twists associated
with these singularities are considered. Viewed from the purely field-theoretic
perspective, such models, which one might call conifold GUTs, require only a
very mild relaxation of the constraints of orbifold model building. Our most
interesting concrete examples include the breaking of E_7 to SU(5) and of E_8
to SU(4)xSU(2)xSU(2) (with extra factor groups), where three generations of
standard model matter come from the gauge sector and the families are
interrelated either by SU(3) R-symmetry or by an SU(3) flavour subgroup of the
original gauge group.
