This work deals with semistably degenerating families of polarized abelian varieties over a scheme. We extend to this case the basic tools classically associated with polarizations -- e.g., the theta group acting on the linear system -- and thus obtain results such as negativity of the determinant sheaf of the relative Lie algebra. Whenever possible we avoid making restrictive hypotheses such as separability of the polarization. The central tool is a formula generalizing the fact that (over $\mathbb C$) theta constants are modular forms ; this is proved (Chapter VIII) when the base is equicharacteristic or when the family does not degenerate.
Finally we give applications to abelian varieties over function fields, including as a special case Zarhin's finiteness theorem when the ground field is finite.