ABSTRACT. These lectures concern (nonlinear) filtering. Very roughly the art of obtaining best estimates for some stochastic time-varying variable x on the basis of observations of another process y. The more concrete object under consideration being a stochastic dynamical system dx=f(x)dt+G(x)dw, where w is Wiener noise, with observations dy=h(x)dt+dv, corrupted by further noise. The subject as presented here involves ideas and techniques from Lie algebra theory, stochastics, differential topology, approximation theory and partial differential equations and has relations with quantum theory and stochastic physics. The lectures are adressed to practitioners in any one of these areas assuming that as a rule they are not experts in the other ones.