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 or a group action, by relating it to the homology of its base and fiber. This spectral sequence arises from a filtration of a complex, allowing for the systematic calculation of derived functors. In essence, the spectral sequence organizes information about the layers of a space, revealing how they interact. It is particularly useful in the study of group cohomology and sheaf cohomology, where it helps to understand the relationships between different algebraic structures associated with topological spaces.