Pulsars have enormous magnetic fields whose energy density
dwarfs the rest mass density of their plasma magnetosphere. In this
regime of a plasma, the particles drop out of the description, leaving a set of
equations for the electromagnetic field alone. This non-linear,
deterministic system is known as force-free electrodynamics, and turns out to
have some beautiful and bizarre features. I will give a pedagogical
introduction to these equations and their role in astrophysics and then discuss
our recent contributions. We have taken a geometric viewpoint, using the
null structure of spacetime to unify previous exact solutions and discover new
ones. We have found non-stationary, non-axisymmetric solutions that
describe the outer magnetosphere of pulsars, including those that are
accelerated or torqued. We have derived the standard cartoon of the
aligned pulsar magnetosphere from an explicit, minimal set of assumptions.
Time permitting, I will also discuss some properties of black hole
magnetospheres: Blandford-Znajek energy extraction, non-scattering force-free
waves, and the no-ingrown-hair theorem.