  We discuss several versions of the Family Signature Theorem: in rational
cohomology using ideas of Meyer, in $KO[\tfrac{1}{2}]$-theory using ideas of
Sullivan, and finally in symmetric $L$-theory using ideas of Ranicki. Employing
recent developments in Grothendieck--Witt theory, we give a quite complete
analysis of the resulting invariants. As an application we prove that the
signature is multiplicative modulo 4 for fibrations of oriented Poincar\'e
complexes, generalising a result of Hambleton, Korzeniewski and Ranicki, and
discuss the multiplicativity of the de Rham invariant.
