A foundational course in numerical analysis. We assume good knowledge of real analysis and calculus, but no previous experience in numerical analysis. We introduce numerical methods for solving non-linear equations, ODEs, optimization problems and integrals. We analyze these in detail, with regard to consistency, accuracy, stability and convergence, theoretically and in practice. Complex physical phenomena, from flows to structural deformations, are often modelled using partial differential equations (PDEs). This course provides an introduction to the numerical solution of PDEs . We begin by describing the general principles of computational modelling, and investigate the properties of PDEs derived from common physical problems. We then describe two popular discretisation approaches, the finite-difference and finite-element methods, along with techniques used for their analysis and practical application. In the last part of the course, we consider time-march methods for unsteady problems, and basic iterative techniques for the solution of large algebraic systems.