The Hochschild-Serre spectral sequence is a mathematical tool used in algebraic topology and homological algebra. It provides a way to compute the homology or cohomology of a space that has a fibration structure, particularly when the space can be viewed as a fiber over a base space with a group action. This spectral sequence arises from a short exact sequence of sheaves or groups and helps relate the cohomology of the total space to that of the base and fiber. It is particularly useful in studying the properties of group actions and fibrations in various mathematical contexts.