Maximal functions are of great interest in Harmonic
Analysis, PDEs, Ergodic theory and other areas of Mathematics and their
study goes back to the time of Hardy and Littlewood. In this talk,
firstly, we shall see how they arise very naturally and how to obtain
useful estimates for maximal functions on euclidean spaces using
covering lemmas. The rest of the talk is dedicated to the analysis of
two typical scenarios: the first one is the study of maximal functions
in the absence of covering arguments, and the second is the study of
maximal functions arising from wave equations.