It is well known that the filtering theory has important applications in both military
and commercial industries. The Kalman–Bucy filter has been used in many areas such as navigational
and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the
Kalman–Bucy filter has limited applicability because of the linearity assumptions of the drift term
and observation term as well as the Gaussian assumption of the initial value. Therefore there has
been an intensive interest in solving the nonlinear filtering problem. The central problem of nonlinear
filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we
shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation algebra method,
direct method, and new algorithm method. The first two methods are relatively easy to implement
in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent
advance in the third method which solves all the nonlinear filtering problems in a real-time manner
in theory.