This chapter represents the core of the book. Building on the general theory introduced in previous chapters, stochastic differential equations (SDEs) are presented as a key mathematical tool for relating the subject of dynamical systems to Wiener noise. The well-posedness of an initial value problem for SDEs is proven, and primary analytical and probabilistic properties of the solutions are presented. SDEs are discussed as dynamical representations of Markov diffusions, and their infinitesimal generators are derived as a large class of partial differential operators. As an application of great interest in applications, the Girsanov theorem is proven. The second part of this chapter is devoted to establishing a link between SDEs and partial differential equations (PDEs). In particular, backward and forward Kolmogorov PDEs are introduced, together with the more general Feynman-Kac formula. Later, Dynkin’s theory is presented for the stochastic representation of solutions to linear deterministic PDEs. An introduction to stochastic stability of equilibria is provided, together with the more general concept of Invariant measures for SDEs, both concepts being fundamental in the analysis of random dynamical systems. The chapter closes with an introduction to Ito-Levy SDEs and their infinitesimal generators.