  Charged particles accelerated by electromagnetic fields emit radiation, which
must, by the conservation of momentum, exert a recoil on the emitting particle.
The force of this recoil, known as radiation reaction, strongly affects the
dynamics of ultrarelativistic electrons in intense electromagnetic fields. Such
environments are found astrophysically, e.g. in neutron star magnetospheres,
and will be created in laser-matter experiments in the next generation of
high-intensity laser facilities. In many of these scenarios, the energy of an
individual photon of the radiation can be comparable to the energy of the
emitting particle, which necessitates modelling not only of radiation reaction,
but quantum radiation reaction. The worldwide development of multi-petawatt
laser systems in large-scale facilities, and the expectation that they will
create focussed electromagnetic fields with unprecedented intensities $>
10^{23}~\mathrm{W}\text{cm}^{-2}$, has motivated renewed interest in these
effects. In this paper I review theoretical and experimental progress towards
understanding radiation reaction, and quantum effects on the same, in
high-intensity laser fields that are probed with ultrarelativistic electron
beams. In particular, we will discuss how analytical and numerical methods give
insight into new kinds of radiation-reaction-induced dynamics, as well as how
the same physics can be explored in experiments at currently existing laser
facilities.
