Owing to the constant interplay between theory and applications, the field of differential equations is one of the most dynamic in mathematics. In this text, Vrabie (mathematics, U. of Iasi) offers a nontraditional approach, focusing on the Cauchy problem but separately addressing existence, uniqueness, approximation, and continuation of the solutions, while giving complete proofs for some of the fundamental results at the core of the discipline. She describes progress made in the field across several centuries, classical facts about approximations and stability of solutions, prime integrals, linear and quasi-linear first-order partial differential equations, conservation laws, and various considerations involved with differential equations at the undergraduate level.