We propose a construction of Kahler and non-Kahler 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 P2s. 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.