Abstract: Maximal operators are central objects of study in harmonic analysis. The most classical example is the Hardy-Littlewood maximal operator (in both its centered and uncentered form) but there are also other interesting ones, for instance the ones associated to a convolution with a smooth kernel. In this introductory talk we will introduce the sub-area of harmonic analysis that studies how the variation of a maximal function behaves with respect to the initial data. We will discuss the classical results in this topic (most of them dealing with the one dimensional case) and later we will talk about some recent results that deal with the higher dimensional case when restricted to radial data. Also we will show some analogues of these problems when the domain of the functions are non-trivial manifolds, in particular a recent result on the d-sphere.