This chapter is intended to provide a general overview of the importance of symmetries in Mathematics and Physics. It contains a general introduction to what is meant by symmetry and gives a brief account of the evolution of the concept of symmetry, from the birth of the group theory with Galois to Lie’s intuition of introducing continuous groups of transformations for studying differential equations. Symmetries are the main ingredient for classifying the various geometries (Erlangen program by Felix Klein) and a fundamental tool for discovering the laws of the nature. In particular, the symmetries at the basis of classical mechanics, as well as the frame-indifference principle in continuum mechanics and thermodynamics, and the Noether theorem relating the continuous symmetries of the Lagrangian action functional to conservation laws, are shortly discussed. Finally, after introducing dimensional analysis, and proving the Buckingham \(\pi \) -theorem, widely used for working with dimensionless quantities, the computation made by Taylor in 1950 (when the data were still classified) of the energy relaxed in the first atomic explosion in the New Mexico (Trinity test of the Manhattan project) is detailed.