  In this review, we provide a concise summary of several important
mathematical results for stochastic travelling waves generated by monostable
and bistable reaction-diffusion stochastic partial differential equations
(SPDEs). In particular, this survey is intended for readers new to the topic
but who have some knowledge in any sub-field of differential equations. The aim
is to bridge different backgrounds and to identify the most important common
principles and techniques currently applied to the analysis of stochastic
travelling wave problems. Monostable and bistable reaction terms are found in
prototypical dissipative travelling wave problems, which have already guided
the deterministic theory. Hence,we expect that these terms are also crucial in
the stochastic setting to understand effects and to develop techniques. The
survey also provides an outlook, suggests some open problems, and points out
connections to results in physics as well as to other active research
directions in SPDEs.
