Optimisation, or mathematical programming, is a fundamental subject within decision science and operations research, in which mathematical decision models are constructed, analysed, and solved. This book's focus lies on providing a basis for the analysis of optimisation models and of candidate optimal solutions, especially for continuous optimisation models. The main part of the mathematical material therefore concerns the analysis and linear algebra that underlie the workings of convexity and duality, and necessary/sufficient local/global optimality conditions for unconstrained and constrained optimisation problems. Natural algorithms are then developed from these optimality conditions, and their most important convergence characteristics are analysed. This book answers many more questions of the form: 'Why/why not?' than 'How?'.This choice of focus is in contrast to books mainly providing numerical guidelines as to how optimisation problems should be solved. We use only elementary mathematics in the development of the book, yet are rigorous throughout. This book provides lecture, exercise and reading material for a first course on continuous optimisation and mathematical programming, geared towards third-year students, and has already been used as such, in the form of lecture notes, for nearly ten years. This book can be used in optimisation courses at any engineering department as well as in mathematics, economics, and business schools. It is a perfect starting book for anyone who wishes to develop his/her understanding of the subject of optimisation, before actually applying it.