  In this paper we discuss various aspects of non-compact models of CFT of the
type: $ \prod_{j=1}^{N_L} {N=2 Liouville theory}_j \otimes \prod_{i=1}^{N_M}
{N=2 minimal model}_i $ and $ \prod_{j=1}^{N_L}{SL(2;R)/U(1) supercoset}_j
\otimes \prod_{i=1}^{N_M} {N=2 minimal model}_i $. These models are related to
each other by T-duality. Such string vacua are expected to describe non-compact
Calabi-Yau compactifications, typically ALE fibrations over (weighted)
projective spaces. We find that when the Liouville ($SL(2;R)/U(1)$) theory is
coupled to minimal models, there exist only (c,c), (a,a) ((c,a), (a,c))-type of
massless states in CY 3 and 4-folds and the theory possesses only complex
(K\"{a}hler) structure deformations. Thus the space-time has the characteristic
feature of a conifold type singularity whose deformation (resolution) is given
by the N=2 Liouville (SL(2;R)/U(1)) theory.
  Spectra of compact BPS D-branes determined from the open string sector are
compared with those of massless moduli. We compute the open string Witten index
and determine intersection numbers of vanishing cycles. We also study non-BPS
branes of the theory that are natural extensions of the ``unstable B-branes''
of the SU(2) WZW model in hep-th/0105038.
