  A para-K\"ahler manifold can be defined as a pseudo-Riemannian manifold
$(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a
parallel field of skew-symmetric endomorphisms with $ K^2 = \mathrm{Id} $ or,
equivalently, as a symplectic manifold $(M,\omega)$ with a bi-Lagrangian
structure $L^\pm$, i.e. two complementary integrable Lagrangian distributions.
  A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an
invariant para-K\"ahler structure $(g,K)$ if and only if it is a covering of
the adjoint orbit $\mathrm{Ad}_Gh$ of a semisimple element $h.$ We give a
description of all invariant para-K\"ahler structures $(g,K)$ on a such
homogeneous manifold. Using a para-complex analogue of basic formulas of
K\"ahler geometry, we prove that any invariant para-complex structure $K$ on $M
= G/H$ defines a unique para-K\"ahler Einstein structure $(g,K)$ with given
non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is
given.
  A survey of recent results on para-complex geometry is included.

Differential Geometry