The solution to the problem of nonlinear filtering may be given either as an estimate of the signal (and ideally some measure of concentration), or as a full posterior distribution.  Similarly, one may evaluate the fidelity of the filter either by its ability to track the signal of its proximity to the posterior filtering distribution.  Hence, the field enjoys a lively symbiosis between probability and control theory, and there are plenty of engineering application which benefit from algorithm advances.  This talk will survey some recent theoretical results involving accurate signal tracking with noise-free (degenerate) dynamics in high-dimensions (infinite, in principle, but say d between 10^3 and 10^8, depending on the size of your application and your computer), and high-fidelity approximation of the filtering distribution in low dimensions (say d between 1 and several 10s).