This survey is intended as a mathematical overview of the monogenic signal theory, its current developments as well as of its applications to various problems. Several approaches to the concept of monogenic signal are given, followed by a short discussion on its advantages and drawbacks. Among those, one highlights the definitions arising from different concepts of Hilbert transforms, as well as the concept of the monogenic signal as a solution to a boundary value problem. In the final part, some recent developments in the field are presented.