This thesis provides a novel approach to the problem of state estimation for discrete-time nonlinear systems in the presence of large model uncertainties. Though classical nonlinear Kalman filters such as the extended Kalman filter (EKF) can handle uncertainties by increasing the value of noise covariances, this is only applicable to systems with small uncertainties. To this end, a discretetime robust extended Kalman filter (REKF) is formulated and applied to examples from the fields of aerospace engineering and signal processing with an emphasis on attitude estimation for small unmanned aerial vehicles (UAVs) and image processing under the influence of atmospheric turbulence. The robust filter is an approximate set-valued state estimator where the Riccati and filter equations are obtained as an approximate solution to a reverse-time optimal control problem defining the set-valued state estimator. The advantages of the REKF over the classical EKF are investigated for examples from the fields aerospace engineering and signal processing where large model uncertainties are introduced. In the case of small UAVs, an alternative attitude estimation algorithm based on the REKF is proposed in the event of gyroscopic failure and the inability of the vehicle to carry redundant sensors due to limited payload capabilities. In the case of image reconstruction under atmospheric turbulence, a robust pixel-wandering (random shifts) scheme is proposed to aid the process of image reconstruction. Also, problems pertaining to platform vibration analysis for aerospace vehicles and a frequency demodulation process in the presence of channel-induced uncertainties is also discussed.