The development of an undergraduate mathematical modelling course currently in its fifth iteration at the United States Military Academy is described. The principal purpose of the course designed for sophomores is to provide students the opportunity to practice model construction in an artistic fashion while demonstrating the utility and universality of elementary mathematics. The resolution of issues such as the determination of course prerequisites, choice of model construction or model selection, number and type of student projects, choice of disciplines in which to model, instructor preparation, text selection and development of supporting textual materials is discussed. After a detailed description of the course, the authors discuss lessons learned and evaluate the course.