This chapter is aimed to offer a glimpse on the majorization theory and the beautiful inequalities associated to it. Introduced by G. H. Hardy, J. E. Littlewood, and G. Pólya (Messenger Math. 58:145–152, (1929), [208]) in 1929, and popularized by their celebrated book on Inequalities (Hardy et al., Inequalities, Cambridge University Press, 1952, [209]), the relation of majorization has attracted along the time a big deal of attention not only from the mathematicians, but also from people working in various other fields such as statistics, economics, physics, signal processing, data mining, etc. Part of this research activity is summarized in the 900 pages of the recent book by A. W. Marshall, I. Olkin, and B. Arnold (Inequalities: theory of majorization and its applications. Springer, New York (2011), [305]).