  We prove the global non-linear stability, without symmetry assumptions, of
slowly rotating charged black holes in de Sitter spacetimes in the context of
the initial value problem for the Einstein-Maxwell equations: If one perturbs
the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole,
then in a neighborhood of the exterior region of the black hole, the metric and
the electromagnetic field decay exponentially fast to their values for a
possibly different member of the KNdS family. This is a continuation of recent
work of the author with Vasy on the stability of the Kerr-de Sitter family for
the Einstein vacuum equations. Our non-linear iteration scheme automatically
finds the final black hole parameters as well as the gauge in which the global
solution exists; we work in a generalized wave coordinate/Lorenz gauge, with
gauge source functions lying in a suitable finite-dimensional space.
  We include a self-contained proof of the linear mode stability of
Reissner-Nordstr\"om-de Sitter black holes, building on work by
Kodama-Ishibashi. In the course of our non-linear stability argument, we also
obtain the first proof of the linear (mode) stability of slowly rotating KNdS
black holes using robust perturbative techniques.
