  Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with
normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the
logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections.
We then show that the Albanese morphism $\alpha$ is a fibration with fibers
being spherical (in particular, rational) varieties. It follows that all
irreducible components of $D$ are nonsingular, and any partial intersection of
them is irreducible. Also, the image of $X$ under the morphism $\sigma$
associated with $- K_X - D$ is a spherical variety, and the irreducible
components of all fibers of $\sigma$ are quasiabelian varieties. Generalizing
the Borel-Remmert structure theorem for homogeneous varieties, we show that the
product morphism $\alpha \times \sigma$ is surjective, and the irreducible
components of its fibers are toric varieties. We reduce the classification of
log homogeneous varieties to a problem concerning automorphism groups of
spherical varieties, that we solve under an additional assumption.
