Over an algebraic variety, vector bundles with a symmetric bilinear form, taking values in a possibly non-trivial line bundle, are of increasing interest in arithmetic geometry. These are the natural objects on which to define generalized trace maps. However, until recently these forms have not enjoyed a theory of cohomological invariants analogous to the Hasse-Witt invariants (when the line bundle is trivial). I will give some arithmetic situations where line bundle-valued forms arise, and I will survey the construction of new invariants for these forms in "mod 4" etale cohomology.