  In this thesis we study aspects of strong-field gravitational lensing by
black holes in general relativity, with a particular focus on the role of
integrability and chaos in geodesic motion. We first investigate binary black
hole shadows using the Majumdar-Papapetrou static binary black hole (or
di-hole) solution. It is shown that the propagation of null geodesics on this
spacetime background is a natural example of chaotic scattering. We demonstrate
that the binary black hole shadows exhibit a self-similar fractal structure
akin to the Cantor set. Next, we use techniques from the field of non-linear
dynamics to quantify these fractal structures in binary black hole shadows.
Using a recently developed numerical algorithm, called the merging method, we
demonstrate that parts of the Majumdar-Papapetrou di-hole shadow may possess
the Wada property. We then study the existence, stability and phenomenology of
circular photon orbits in stationary axisymmetric four-dimensional spacetimes.
We employ a Hamiltonian formalism to describe the null geodesics of the
Weyl-Lewis-Papapetrou geometry. Using the Einstein-Maxwell equations, we
demonstrate that generic stable photon orbits are forbidden in pure vacuum, but
may arise in electrovacuum. Finally, we apply a higher-order geometric optics
formalism to describe the propagation of electromagnetic waves on Kerr
spacetime. Using the symmetries of Kerr spacetime, we construct a complex null
tetrad which is parallel-propagated along null geodesics; we introduce a system
of transport equations to calculate certain Newman-Penrose quantities along
rays; we derive generalised power series solutions to these transport equations
through sub-leading order in the neighbourhood of caustic points; and we
introduce a practical method to evolve the transport equations beyond caustic
points.
