  We give a construction of integrable complex structures on the total space of
a smooth principal bundle over a complex manifold, with an even dimensional
compact Lie group as structure group, under certain conditions. This
generalizes the constructions of complex structure on compact Lie groups by
Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others.
It also yields large classes of new examples of non-K\"ahler compact complex
manifolds. Moreover, under suitable restrictions on the base manifold, the
structure group, and characteristic classes, the total space of the principal
bundle admits SKT metrics. This generalizes recent results of Grantcharov et
al. We study the Picard group and the algebraic dimension of the total space in
some cases. We also use a slightly generalized version of the construction to
obtain (non-K\"ahler) complex structures on tangential frame bundles of complex
orbifolds.
