A lower and upper bound approach on the optimal mean square error is used to study the asymptotic behavior of one dimensional nonlinear filters. Two aspects are treated: (1) The long time behavior (t --> INFINITY). (2) The asymptotic behavior as a small parameter EPSILON-->0. Lower and upper bounds that satisfy Riccati equations are derived and it is shown that for nonlinear systems with linear limiting systems, the Kalman filter designed for the limiting systems is asymptotically optimal in a reasonable sense. In the case of nonlinear systems with low measurement noise level, three asymptotically optimal filters are provided one of which is linear. In chapter 4, the stationary behavior of the Benes filter is investigated.