  We consider bilinear pseudo-differential operators whose symbols posses
Gevrey type regularity and may have a sub-exponential growth at infinity,
together with all their derivatives. It is proved that those symbol classes can
be described by the means of the short-time Fourier transform and modulation
spaces. Our first main result is the invariance property of the corresponding
bilinear operators. Furthermore we prove the continuity of such operators when
acting on modulation spaces. As a consequence, we derive their continuity on
anisotropic Gelfand-Shilov type spaces. We consider both Beurling and Roumieu
type symbol classes and Gelfand-Shilov spaces.
