Providing an introduction to the qualitative theory of ordinary differential equations, this text also shows how the theory may be applied to dynamical systems not only in mechanics and electricity but also in ecology, economics and many other spheres. The mathematical ideas involved consist of a blend of calculus, algebra and geometry so that the reader makes use of a variety of mathematical skills. Aimed at second-year undergraduates who are assumed to have completed a first course in calculus of several variables and linear algebra, the authors have provided, wherever possible, examples to explain the significance of the theorems introduced.