  Let G be a reductive algebraic group over the complex number filed, and K =
G^{\theta} be the fixed points of an involutive automorphism \theta of G so
that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K
respectively and consider a product of partial flag varieties G/P and K/Q with
diagonal K-action. The double flag variety G/P \times K/Q thus obtained is said
to be of finite type if there are finitely many K-orbits on it. A triple flag
variety G/P^1 \times G/P^2 \times G/P^3 is a special case of our double flag
varieties, and there are many interesting works on the triple flag varieties.
In this paper, we study double flag varieties G/P \times K/Q of finite type. We
give efficient criterion under which the double flag variety is of finite type.
The finiteness of orbits is strongly related to spherical actions of G or K.
For example, we show a partial flag variety G/P is K-spherical if a product of
partial flag varieties G/P \times G/\theta(P) is G-spherical. We also give many
examples of the double flag varieties of finite type, and for type AIII, we
give a classification when P = B is a Borel subgroup of G.
