  We introduce and study a new notion of stability for varieties fibered over
curves, motivated by Koll\'ar's stability for homogeneous polynomials with
integral coefficients. We develop tools to study geometric properties of stable
birational models of fibrations whose fibers are complete intersections in
weighted projective spaces. As an application, we prove the existence of
standard models of threefold degree one and two del Pezzo fibrations, settling
a conjecture of Corti from 1996.
