In this chapter, simple Gaussian and non-Gaussian SDEs are presented. They are appropriate paradigms to understand key dynamical and statistical features of general complex systems. They also serve as simple illustrations for introducing many commonly used mathematical tools for analyzing more sophisticated systems. Important concepts, such as equilibrium statistics, decorrelation time, and additive versus multiplicative noise, are discussed. Reynolds decomposition is also introduced, which can be combined with Itô’s formula to solve the time evolution of moment equations. In addition, the study of the path-wise behavior of linear systems is presented via the application of Itô’s formula. Similarly, the equilibrium statistics of simple non-Gaussian systems can be found analytically by solving the stationary Fokker-Planck equation. To facilitate the study of the statistical behavior of complex nonlinear systems, closure methods are introduced, which are useful tools for approximating the moment equations in the presence of strong nonlinearity. Finally, a family of nonlinear SDEs with exactly solvable conditional moments is developed, aiming at providing a powerful tool to approximate complex nonlinear and non-Gaussian systems by a suitable coarse-graining of the conditional statistics.