This a cohomology theory for smooth manifolds, where the (co)chain complex is defined by differential forms on a smooth manifold with differential given by exterior derivative. Then $n^{th}$ de Rham cohomology group is the quotient "closed $n$-forms/exact $n$-forms". Use in conjunction with other algebraic topology and differential geometry related tags if necessary.

Given a smooth $n$-dimensional manifold $M$, there is a cochain complex

$$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \dots \xrightarrow{d} \Omega^{n-1}(M) \xrightarrow{d} \Omega^n(M) \to 0$$

of differential forms with exterior derivative as the differential, called the *de Rham complex* (named after Georges de Rham). The cohomology of this complex is called *de Rham cohomology*: $$H^k_{\text{dR}}(M) = H^k(\Omega^{\bullet}(M), d)$$

These quotient abelian groups (in fact, real vector spaces) measures the extent to which closed $k$-forms to be exact. As a consequence of Hodge theorem if $M$ is compact, $H^k_{\text{dR}}(M)$ is a finite-dimensional vector space for every $k$. Also, by Poincaré lemma, every closed differential form is locally exact and therefore contractible spaces have trivial de Rham cohomology.

By Stokes' theorem, integration of differential forms along singular chains induces, for any compact smooth manifold $M,$ a bilinear pairing

$$H^k_{\text{Sing}}(M)\times H^k_{\text{dR}}(M)\to\mathbb{R}$$

de Rham's theorem asserts that this pairing induces an isomorphism between singular cohomology with real coefficients and de Rham cohomology by showing each vector space in above pairing is dual to one another. Moreover, it coincides with the Čech cohomology.