The credo ‘every effect in nature follows a maximum or minimum rule’ is what Euler wrote in 1744 and which since then has driven the development of the mathematical field of Calculus of Variations. We will give an introduction into this area focusing both on problems from pure mathematics and from applications in atomic physics. Symmetries will play a major role in this analysis. Topics include the isoperimetric inequality, the concentration compactness principle, rearrangements and the moving plane method, Sobolev and Lieb-Thirring inequalities and the semi-classical limit of the Schrödinger equation.