This course introduces students to the modern theory and methods of ordinary and partial differential equations. It builds on the classical foundations of differential equations studied in second year. Ordinary and partial differential equations form an essential part of the mathematical background required for engineering and the physical sciences. A large number of real-life problems can be modelled using differential equations, making the subject one of the most widely applicable areas of mathematics. The course concentrates on some fundamental analytical and numerical methods for applied differential equations arising from the mathematical modelling of physical, chemical and biological systems.