This chapter is exclusively dedicated to the introduction and study of the fundamental concepts and results concerning the main topic of this book: the so-called Cauchy problem, or the initial-value problem. In the first section we define the Cauchy problem for a given differential equation and the basic concepts referring to: local solution, saturated solution, global solution, etc. In the second section we prove that a sufficient condition in order that a Cauchy problem have at least one local solution is the continuity of the function f. In the third one we present several specific situations in which every two solutions of a certain Cauchy problem coincide on the common part of their domains. The existence of saturated solutions as well as of global solutions is studied in the fourth section. In the fifth section we prove several results concerning the continuous dependence of the saturated solutions on the initial data and on the parameters, while in the sixth one we discuss the differentiability of saturated solutions with respect to the data and to the parameters. The seventh section reconsiders all the problems previously studied in the case of the n ^th-order scalar differential equation. The last section contains several exercises and problems illustrating the most delicate aspects of the abstract theory.