  In the nearly seven decades since the publication of Alan Turing's work on
morphogenesis, enormous progress has been made in understanding both the
mathematical and biological aspects of his proposed reaction-diffusion theory.
Some of these developments were nascent in Turing's paper, and others have been
due to new insights from modern mathematical techniques, advances in numerical
simulations, and extensive biological experiments. Despite such progress, there
are still important gaps between theory and experiment, with many examples of
biological patterning where the underlying mechanisms are still unclear. Here
we review modern developments in the mathematical theory pioneered by Turing,
showing how his approach has been generalized to a range of settings beyond the
classical two-species reaction-diffusion framework, including evolving and
complex manifolds, systems heterogeneous in space and time, and more general
reaction-transport equations. While substantial progress has been made in
understanding these more complicated models, there are many remaining
challenges that we highlight throughout. We focus on the mathematical theory,
and in particular linear stability analysis of `trivial' base states. We
emphasise important open questions in developing this theory further, and
discuss obstacles in using these techniques to understand biological reality.
