Abstract: Studying maps between manifolds is a self-motivating topic of research, as it is fundamental and general. Moreover, interesting properties arise when one provides an analytic structure to some object, so it is a good class of maps to study. Finally, in some cases, it is convenient to follow the mindset of divide et impera and divide a question into simpler problems to conquer some mathematical mysteries. This leads us to study holomorphic map germs. We will see basic concepts of this theory; similarities between this and other theories and developments, as the real global case that Thom and Mather studied or hypersurfaces with isolated singularities; one of my fields of research, germs from C^n to C^p with n < p; and some fundamental (in the sense of dealing with fundamental objects) theorems my PhD advisor, J.J. Nuno-Ballesteros, and me have attained.