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A theorem expressing the real cohomology groups of a differentiable manifold  $  M $
in terms of the complex of differential forms (cf. [[Differential form|Differential form]]) on  $  M $. 
If  $  E  ^ {*} ( M) = \sum _ {p = 0 }   ^ {n} E  ^ {p} ( M) $
is the de Rham complex of  $  M $, 
where  $  E  ^ {p} ( M) $
is the space of all infinitely-differentiable  $  p $-
forms on  $  M $
equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra  $  H  ^ {*} ( E  ^ {*} ( M)) $
of the complex  $  E  ^ {*} ( M) $
and the cohomology algebra  $  H  ^ {*} ( M, \mathbf R ) $
of  $  M $
with values in  $  \mathbf R $. 
An explicit interpretation of this isomorphism is that to each closed  $  p $-
form  $  \omega $
there is associated a linear form  $  \gamma \rightarrow \int _  \gamma  \omega $
on the space of  $  p $-
dimensional singular cycles  $  \gamma $
in  $  M $.

The theorem was first established by G. de Rham [[#References|[1]]], although the idea of a connection between cohomology and differential forms goes back to H. Poincaré.

There are various versions of de Rham's theorem. For example, the cohomology algebra  $  H  ^ {*} ( E _ {c}  ^ {*} ( M)) $
of the complex  $  E _ {c}  ^ {*} ( M) $
of forms with compact supports is isomorphic to the real cohomology algebra  $  H _ {c}  ^ {*} ( M, \mathbf R ) $
of the manifold  $  M $
with compact supports. The cohomology of  $  M $
with values in a locally constant sheaf of vector spaces is isomorphic to the cohomology of the complex of differential forms with values in the corresponding flat vector bundle [[#References|[3]]]. The cohomology of a simplicial set  $  S $
with values in any field  $  k $
of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over  $  k $. 
In the case when  $  S $
is the singular complex of an arbitrary topological space  $  X $
one obtains in this way a graded-commutative differential graded  $  k $-
algebra  $  A _ { \mathop{\rm dR}  } ( X) $
with cohomology algebra  $  H  ^ {*} ( A _ { \mathop{\rm dR}  } ( X)) $
isomorphic to the singular cohomology algebra  $  H  ^ {*} ( X, k) $(
see [[#References|[4]]]). If  $  X $
is a smooth affine algebraic variety over  $  \mathbf C $, 
then the cohomology algebra  $  H  ^ {*} ( X, \mathbf C ) $
is isomorphic to the cohomology algebra of the complex of regular differential forms on  $  M $(
see [[De Rham cohomology|de Rham cohomology]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. de Rham,   "Sur l'analysis situs des variétés à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033037.png" /> dimensions"  ''J. Math. Pures Appl. Sér. 9'' , '''10'''  (1931)  pp. 115–200</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. de Rham,   "Differentiable manifolds" , Springer  (1984)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.S. Raghunathan,   "Discrete subgroups of Lie groups" , Springer  (1972)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Lehmann,   "Théorie homotopique des forms différentiélles (d'après D. Sullivan)"  ''Astérisque'' , '''45'''  (1977)</TD></TR></table>