Stochastic modelling via stochastic differential equations (SDE) plays an important role in science and applied science. Professor Mao’s book reflects the need of those working in mathematical modelling for a deep but understandable presentation not only of the classical Itô SDE theory (functional and backward equations, and equations of neutral type) but also for a summary of new developments, primarily the Carathéodory and the Cauchy-Marayama approximation procedures, in addition to the classical Picard construction. Approximations are used both to prove existence and uniqueness of solutions and to obtain precise numerical solutions needed in applications. The text also offers an analysis of stability in stochastic modelling. The general Lyapunov method and Razumkin calculus are applied in a study of exponential stability and asymptotic bounds in the SDE environment. An illustration of the modelling by means of stochastic differential equations covers stochastic oscillators, stochastic finance and stochastic neural networks. The book makes SDE theory accessible to beginners in the field and to a wide range of scientists and engineers without forcing the reader through a jungle of mathematical details.