Differential equations can be viewed as a tool well suited to bring mathematics closer to real life and describe phenomena with an origin in physics, chemistry, biology, economics, etc. Of course, this 'invites' scientists and engineers to use them.On the other hand, differential equations are at the starting point of a lot of purely mathematical activity. Thus, a large part of functional analysis has been motivated by the need to solve questions coming from the analysis of differential systems. The same can be said of numerical analysis. Also, we must remember the relevant role played by differential equations in the formulation and resolution of problems in harmonic analysis, differential geometry, probability calculus, etc.Consequently, the interest in differential equations is justified from a double viewpoint: as significative and inspiring examples of questions and results arising in many areas in mathematics and also as a machinery that must be well understood in order to apply mathematics to real world problems.The main goal of this book is to provide a rigorous introduction to the theoretical study of this important tool and, also, to demonstrate its utility with applications coming from many fields.The book has its origin in several 'classical' or 'standard' courses given by the author for decades at the University of Sevilla. It has been an objective to make compatible the rigor of the underlying mathematical theory and the richness of applicability. Thus, together with existence, uniqueness, regularity, continuous dependence on data and parameters, etc., permanent interpretation of the laws, the role of the data, the behaviour of the solutions and other elements are discussed. Although briefly, the connections to first order partial differential equations and control and optimization have also been indicated.Each chapter has been completed with a collection of exercises, many of them with useful hints.