  We study aspects of superstring vacua of non-compact special holonomy
manifolds with conical singularities constructed systematically using soluble N
= 1 superconformal field theories (SCFT's). It is known that Einstein
homogeneous spaces G/H generate Ricci flat manifolds with special holonomies on
their cones R_+ x G/H, when they are endowed with appropriate geometrical
structures, namely, the Sasaki-Einstein, tri-Sasakian, nearly Kahler, and weak
G_2 structures for SU(n), Sp(n), G_2, and Spin(7) holonomies, respectively.
Motivated by this fact, we consider the string vacua of the type: R^{d-1,1} x
(N = 1 Liouville) x (N=1 supercoset CFT on G/H) where we use the affine Lie
algebras of G and H in order to capture the geometry associated to an Einstein
homogeneous space G/H. Remarkably, we find the same number of spacetime and
worldsheet SUSY's in our ``CFT cone'' construction as expected from the
analysis of geometrical cones over G/H in many examples. We also present an
analysis on the possible Liouville potential terms (cosmological constant type
operators) which provide the marginal deformations resolving the conical
singularities.
