Abstract: The Cartan-Serre-Grothendieck theorem implies that all the cohomological subtleties associated to a coherent sheaf on a projective space disappear after twisting by a sufficiently high multiple of the hyperplane line bundle. Castelnuovo-Mumford regularity gives a quantitative measure of how much one has to twist in order that these properties take effect. It then governs the algebraic complexity of a coherent sheaf. In this talk, we give the definition and basic properties of Castelnuovo-Mumford regularity, and introduce some applications.