  We propose a construction of K\"ahler and non-K\"ahler Calabi-Yau manifolds
by branched double covers of twistor spaces. In this construction we use the
twistor spaces of four-manifolds with self-dual conformal structures, with the
examples of connected sum of $n$ $\mathbb{P}^{2}$s. We also construct
$K3$-fibered Calabi-Yau manifolds from the branched double covers of the
blow-ups of the twistor spaces. These manifolds can be used in heterotic string
compactifications to four dimensions. We also construct stable and polystable
vector bundles. Some classes of these vector bundles can give rise to
supersymmetric grand unified models with three generations of quarks and
leptons in four dimensions.
