The aim of this paper is the study of functional differential equations with discontinuous right-hand side. In order to implement the fundamental idea of Filippov's theory and to define an analog of a solution in the Filippov sense, the authors suggest a formal procedure of obtaining a functional differential inclusion from a general functional differential equation with discontinuous right-hand side. Next, the authors give sufficient conditions to prove local existence, uniqueness of the (Filippov) solutions and their dependence on parameters. Also, the existence of global solutions is given. This theory is applied to the analysis of gene regulatory networks with general delays.