In this chapter, we review the most fundamental phenomena in differentiable, finite-dimensional multiple-time scale systems as well as the geometric singular perturbation theory (GSPT) that can be used to analyze such systems. We put special emphasis on loss of normal hyperbolicity and illustrate how it relates to the different phenomena. Moreover, by working on lower-dimensional model problems, we review the main technical tool of GSPT, the blowup method, and demonstrate - in line with recent research in this area - how it may provide detailed insight into problems of this kind.