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The cup product is a bilinear operation in cohomology giving it a ring structure. It combines elements from two different cohomology groups to form a new element in a higher-degree cohomology group.

Cohomology rings are graded rings, meaning they are direct sums of cohomology groups of different degrees, with the cup product respecting this grading.

Poincaré Duality relates the $k$th cohomology group to the $(n-k)$th cohomology group in an $n$-dimensional closed orientable manifold, often leading to symmetry in cohomology rings.

Cohomology rings are universal for contravariant functors from the category of topological spaces to the category of graded rings that convert homotopy equivalences into isomorphisms.

A continuous map between spaces $f: X \to Y$ induces a ring homomorphism $f^*: H^*(Y) \to H^*(X)$ on cohomology rings, respecting the cup product.

The cohomological dimension of a space is the largest integer $n$ such that $H^n(X;R) \neq 0$ for a coefficient ring $R$. It provides a measure of the 'size' of a space in terms of cohomology.

Cohomology can be computed with local coefficients, which can capture local twisting of the topology that can't be seen with ordinary (constant) coefficients.

Characteristic classes are cohomology classes that provide algebraic invariants of vector bundles. They play an important role in classifying and understanding bundles over topological spaces.

A tool for computing the cohomology of a fibration. The spectral sequence's $E_2$ page often incorporates simpler cohomology rings and converges to the cohomology of the total space of a fibration.