Weight decompositions were first introduced in the setting of rational homotopy as a tool to study $p$-universal spaces. The same notion may be adapted to many other algebraic contexts and, in general, positive and pure weight decompositions have strong homotopical consequences, often related to formality. A main source of weights is algebraic geometry, either via the theory of mixed Hodge structures on de Rham cohomology or the theory of Galois actions in étale cohomology.