The main goal of this chapter is to present several methods to approach some Cauchy problems which, from various reasons, do not find their place in the preceding theoretical framework. In order to extend the concept of solution in the case of linear differential equations and systems with discontinuous right-hand sides, in the first three sections we introduce and study the notion of distribution as a generalization of an infinitely many differentiable function. In the same spirit, in the fourth section, we present another type of solution suitable for the nonlinear case when the function f on the right-hand side is discontinuous with respect to the t variable. In the next two sections, we discuss two variants of approaching some Cauchy problems for which f is discontinuous as a function of the state variable x, situation involving much more difficulties than the preceding one. In both cases, the manner of approach consists in replacing the differential equation with a so-called differential inclusion. The sixth section is concerned with the study of a class of variational inequalities, while in the next four sections we deal with a Cauchy problem in which the function on the right-hand side of the equation is defined on a set which is not open. In the eleventh section, we present an existence and uniqueness result referring to the Cauchy problem for a class of systems of first-order nonlinear partial differential equations of type gradient. The chapter ends with a section of Exercises...