A length function l on a group G is a (nonnegative) real-valued function that is conjugate invariant, homogenous, and subadditive for commuting pairs. Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on CAT(0) spaces and Gromov delta-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, smooth entropy, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings, and Cremona groups. As applications, we prove rigidity results on group homomorphisms.