This chapter discusses the Gaussian assumption, and a solution to obtain a priori and a posteriori pdf for nonlinear and non-Gaussian system. When the nonlinear stochastic systems are linear and the concerned noise pdfs are assumed to be Gaussian, the closed form solution of the posterior pdf exists in the minimum mean square error sense, which is known as the Kalman filter. The chapter examines the performance of the Gaussian sum (GS) and adaptive GS filters where different sigma point filters are used as a proposal. It describes the concept of representing an arbitrary probability density function with the help of a weighted sum of many Gaussian distributions. This concept has been applied to state estimation problems which leads to a nonlinear estimation method known as a GS filter. Any nonlinear filtering heuristic can be used as a proposal of the GS filter.