  The purpose of this review is to present a comprehensive overview of the
theory of ensemble Kalman-Bucy filtering for continuous-time, linear-Gaussian
signal and observation models. We present a system of equations that describe
the flow of individual particles and the flow of the sample covariance and the
sample mean in continuous-time ensemble filtering. We consider these equations
and their characteristics in a number of popular ensemble Kalman filtering
variants. Given these equations, we study their asymptotic convergence to the
optimal Bayesian filter. We also study in detail some non-asymptotic
time-uniform fluctuation, stability, and contraction results on the sample
covariance and sample mean (or sample error track). We focus on testable
signal/observation model conditions, and we accommodate fully unstable (latent)
signal models. We discuss the relevance and importance of these results in
characterising the filter's behaviour, e.g. it's signal tracking performance,
and we contrast these results with those in classical studies of stability in
Kalman-Bucy filtering.We also provide a novel (and negative) result proving
that the bootstrap particle filter cannot track even the most basic unstable
latent signal, in contrast with the ensemble Kalman filter (and the optimal
filter). We provide intuition for how the main results extend to nonlinear
signal models and comment on their consequence on some typical filter
behaviours seen in practice, e.g. catastrophic divergence.
