It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai ( DMZ) equation in real time and in a memoryless manner. The purpose of this paper is to show that, under very mild conditions ( which essentially say that the growth of the observation vertical bar h vertical bar is greater than the growth of the drift vertical bar f vertical bar), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution u(R) of the DMZ equation on the ball B(R) with u(R)vertical bar(partial derivative BR) = 0. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution uR can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on BR. In theory, our algorithm can solve basically all engineering problems in real time. Specifically, we show that the solution obtained from our algorithms converges to the solution of the DMZ equation in the L(1) sense. Equally important, we have a precise error estimate of this convergence, which is important in numerical computation.