\begin{definition}[Definition:De Rham Cohomology]
Let $X$ be a smooth manifold.
Let $\struct {\Omega^\bullet, d}$ be the de Rham complex of $X$.
Then the '''de Rham Cohomology''' of $X$ is defined to be the cohomology groups of $\Omega^\bullet$.
{{explain|Pick the notation that will be used on {{ProofWiki}} and relegate the rest to an Also denoted as section. Let's try and pick something that's minimally unwieldy.}}
It is usually denoted by $\map {H^\bullet_{\text {de Rham} } } X$, $\map {H^\bullet_{\text {dR}} } X$ or when there is no ambiguity, $\map {H^\bullet} X$.
\end{definition}