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. In this talk, I will review such weight structures defined at the cochain level, together with some main homotopical implications related to formality over the rationals and also over finite fields. This is mostly joint work with Geoffroy Horel and some work in progress with Bashar Saleh.

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