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.
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