Stochastic modelling provides powerful tools for investigating complex phenomena by allowing for many degrees of freedom of a system to be taken into account in a statistical sense.  This can greatly simplify descriptions and analysis by requiring only investigation of a few key parameters describing statistical features of the system and for the investigation of relatively few central degrees of freedom.  Such approaches are at the foundations of statistical mechanics and the historic basis for the development of the field of stochastic processes.   In this lecture we shall give a self-contained survey of the field of stochastic analysis.  By way of several motivating examples we shall develop the theory of Stochastic Differential Equations (SDEs) and Ito Calculus (IC).  We will also establish several useful connections between SDEs and PDEs.  In practice, many stochastic models of interest exhibit dynamics spanning a wide range of time-scales for which the slowest time-scales are often of the most interest.  We shall discuss systematic averaging procedures to obtain reduced stochastic descriptions in terms of a few number of so-called "slow degrees of freedom" obtained from the full description of the stochastic dynamics.  Another important approach to investigate stochastic models in practice is through the use of numerical methods.  We shall discuss in detail several approaches for approximating SDEs and corresponding approaches to develop computational methods.  Throughout, we shall discuss practical applications of the above approaches to problems arising in the sciences.