  We study deformation theory of elliptic fibre bundles over curves in positive
characteristics. As applications, we give examples of non-liftable elliptic
surfaces in charactertic two and three, which answers a question of Katsura and
Ueno. Also, we construct a class of elliptic fibrations, whose liftability is
equivalent to a conjecture of F. Oort concerning the liftability of
automorphisms of curves. Finally, we classify deformations of bielliptic
surfaces.
