I will introduce the concepts of backward, forward, and covariant Lyapunov vectors (that are basically “eigenfunctions” of certain operators) for a dynamical system and discuss their relevance in studying various stability results, both for the system itself and for nonlinear filtering problem, which will also be introduced along the way. The “relevance” to nonlinear filters is in the form of open questions whose answers (as a set of conjectures) may be gleaned from numerical results and linear filtering theory.