  Data assimilation addresses the general problem of how to combine model-based
predictions with partial and noisy observations of the process in an optimal
manner. This survey focuses on sequential data assimilation techniques using
probabilistic particle-based algorithms. In addition to surveying recent
developments for discrete- and continuous-time data assimilation, both in terms
of mathematical foundations and algorithmic implementations, we also provide a
unifying framework from the perspective of coupling of measures, and
Schr\"odinger's boundary value problem for stochastic processes in particular.
