  We study the moduli spaces which classify smooth surfaces along with a
complex line bundle. There are homological stability and Madsen--Weiss type
results for these spaces (mostly due to Cohen and Madsen), and we discuss the
cohomological calculations which may be deduced from them. We then relate these
spaces to (a generalisation of) Kawazumi's extended mapping class groups, and
hence deduce cohomological information about these.
  Finally, we relate these results to complex algebraic geometry. We construct
a holomorphic stack classifying families of Riemann surfaces equipped with a
fibrewise holomorphic line bundle, which is a gerbe over the universal Picard
variety, and compute its holomorphic Picard group.
