The asymptotic behavior as a small parameter EPSILON --> 0 is investigated for one dimensional nonlinear filtering problems. Both weakly nonlinear systems (WNL) and systems measured through a low noise channel are considered. Upper and lower bounds on the optimal mean square error combined with perturbation methods are used to show that, in the case of WNL, the Kalman filter formally designed for the underlying linear systems is asymptotically optimal in some sense. In the case of systems with low measurement noise, three asymptotically optimal filters are provided, one of which is linear. Examples with simulation results are provided.