The aim of the course is to introduce the concepts, rationale, and tools needed to derive and analyse mathematical models of physical phenomena. Mathematical modelling takes place in two basic stages. The first arises from the need of extracting a simpler mathematical description from a physical theory which is intractable in its full generality. This usually targets a specific mechanism believed to be the most relevant in a certain problem to be solved. At this stage physical insight into the phenomena to be studied is crucial. Once the mathematical model has been defined, there remains the task of extracting useful information through the analysis of its solutions. This stage necessarily consists of a rather broad collection of tools and techniques, in classical as well as modern mathematics, rather than a self-consistent theory founded on a limited set of fundamental assumptions. Because of this, mathematical modelling is best learned by first working within the context of specific examples. However, inevitably one finds that the experience gained within a particular application does not quite transfer to a new problem to be solved. After studying in detail a core set of examples, the course will therefore emphasize unifying ideas and stress their theoretical background. These ideas will then be tested on a set of problems close to the current research interests of the instructor. While the necessary physical background for all problems will be developed from scratch in class and through independent reading, some familiarity with the fundamentals of Fluid Mechanics is assumed. The prerequisite mathematical tools will be at the level of those developed in the Methods of Applied Mathematics two-semester sequence. Some more advanced tools will be developed during the second part of the course. Exposure to analysis of ordinary and partial differential equations and numerical methods for scientific computing is recommended.