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An ordinary [[Equivariant cohomology|equivariant cohomology]] for a finite group  $  G $, 
defined in [[#References|[a1]]], on the [[Category|category]]  $  G $-
CW of  $  G $-
complexes (cf. [[Complex|Complex]]; [[CW-complex|CW-complex]]). The objects of  $  G $-
CW are the CW-complexes  $  X $
with a cellular action of  $  G $, 
satisfying the condition that, for every subgroup  $  H $
of  $  G $, 
the fixed point set  $  X  ^ {H} $
is a subcomplex of  $  X $. 
The morphisms are the cellular  $  G $-
mappings. Let  $  O _ {G} $
be the full subcategory of  $  G $-
CW whose objects are the  $  G $-
orbits  $  G/H $, 
where  $  H $
is a subgroup of  $  G $. 
For every contravariant [[Functor|functor]]  $  M $
from  $  O _ {G} $
to the category  $  { \mathop{\rm Ab} } $
of Abelian groups, there exists a Bredon cohomology theory  $  \{ H  ^ {n} _ {G} ( -,M ) \} _ {n \in \mathbf N }  $
which, after restriction to  $  O _ {G} $, 
vanishes for  $  n > 0 $
and is equal to  $  M $
for  $  n = 0 $.

Let  $  c _ {*} ( X ) $
be the chain [[Complex|complex]] of functors from  $  O _ {G} $
to  $  { \mathop{\rm Ab} } $
such that, for every subgroup  $  H $
of  $  G $, 
$  c _ {*} ( X ) ( G/H ) = C _ {*} ( X  ^ {H} ) $
is the ordinary cellular chain complex of  $  X  ^ {H} $. 
Then

$$ 
H  ^ {n} _ {G} ( X,M ) = H  ^ {n} ( { \mathop{\rm Hom} } _ {O _ {G}  } ( c _ {*} ( X ) ,M ) ) ,
$$

where  $  { \mathop{\rm Hom} } _ {O _ {G}  } ( -, - ) $
denotes the set of natural transformations of functors. The functors  $  c _ {n} ( X ) $
are projective objects in the [[Category|category]] of coefficient systems and there is a [[Spectral sequence|spectral sequence]]

$$ 
{ \mathop{\rm Ext} }  ^ {p} _ {O _ {G}  } ( h _ {q} ( X ) ,M ) \Rightarrow H ^ {p + q } _ {G} ( X,M ) ,
$$

where  $  h _ {q} ( X ) ( G/H ) = H _ {q} ( X  ^ {H} ) $.

Let  $  Y $
be a  $  G $-
space with base point  $  y _ {0} $(
cf., e.g., [[Equivariant cohomology|Equivariant cohomology]]). Important examples of coefficient systems are the [[Homotopy|homotopy]] group functors  $  \pi _ {n} ( Y ) $
defined by  $  \pi _ {n} ( Y ) ( G/H ) = \pi _ {n} ( Y  ^ {H} ,y _ {0} ) $. 
The [[Obstruction|obstruction]] theory for  $  G $-
mappings  $  f : X \rightarrow Y $
is formulated in terms of the cohomology groups  $  H  ^ {n} _ {G} ( X, \pi _ {m} ( Y ) ) $. 
For any coefficient system  $  M $
and natural number  $  n > 0 $, 
there is a pointed Eilenberg–MacLane  $  G $-
complex  $  K ( M,n ) $
such that  $  \pi _ {n} ( K ( M,n ) ) = M $
and  $  \pi _ {m} ( K ( M,n ) ) $
vanishes whenever  $  n \neq m $. 
For every  $  G $-
complex  $  X $, 
$  H  ^ {n} _ {G} ( X,M ) = [ X,K ( M,n ) ] _ {G} $, 
where  $  [ -, - ] _ {G} $
denotes  $  G $-
homotopy classes of  $  G $-
mappings.

If  $  h  ^ {*} _ {G} ( - ) $
is an equivariant cohomology theory defined on the category  $  G $-
CW, then there exists an Atiyah–Hirzebruch-type spectral sequence

$$ 
H  ^ {p} _ {G} ( X,h  ^ {q} ) \rightarrow h ^ {p + q } _ {G} ( X ) ,
$$

where  $  h  ^ {q} $
is the restriction of  $  h  ^ {q} _ {G} ( - ) $
to  $  O _ {G} $. 
Bredon cohomology for an arbitrary topological group is studied in [[#References|[a4]]] and [[#References|[a5]]]. Singular ordinary equivariant cohomology is defined in [[#References|[a2]]] (the finite case) and in [[#References|[a3]]]. If a coefficient system  $  M $
is a Mackey functor, then the Bredon cohomology  $  H  ^ {*} _ {G} ( -,M ) $
can be extended to an ordinary  $  { \mathop{\rm RO} } ( G ) $-
graded cohomology [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E. Bredon,   "Equivariant cohomology theories" , ''Lecture Notes in Mathematics'' , '''34''' , Springer  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Bröcker,   "Singuläre Definition der äquivarianten Bredon Homologie"  ''Manuscr. Math.'' , '''5'''  (1971)  pp. 91–102</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Illman,   "Equivariant singular homology and cohomology" , ''Memoirs'' , '''156''' , Amer. Math. Soc.  (1975)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  T. Matumoto,   "Equivariant cohomology theories on G-CW-complexes"  ''Osaka J. Math.'' , '''10'''  (1973)  pp. 51–68</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S.J. Wilson,   "Equivariant homology theories on G-complexes"  ''Trans. Amer. Math. Soc.'' , '''212'''  (1975)  pp. 155–171</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.G. Lewis,   J.P. May,   J. McClure,   "Ordinary RO(G)-graded cohomology"  ''Bull. Amer. Math. Soc.'' , '''4'''  (1981)  pp. 208–212</TD></TR></table>