In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709–769 to study the unipotent section conjecture. The non-abelian cohomology
H
n
a
b
1
(
G
,
P
)
H^1_{\mathrm {nab}}(\mathcal {G},\mathcal {P})
is a scheme. The argument
G
\mathcal {G}
is a proalgebraic group; the coefficient group
P
\mathcal {P}
is prounipotent with trivial center and endowed with an outer action of
G
\mathcal {G}
. This outer action uniquely determines an extension
G
^
\widehat {\mathcal {G}}
of
G
\mathcal {G}
by
P
\mathcal {P}
. With suitable hypotheses, the scheme
H
n
a
b
1
(
G
,
P
)
H^1_{\mathrm {nab}} (\mathcal {G},\mathcal {P})
parametrizes the
P
\mathcal {P}
conjugacy classes of sections of
G
^
→
G
\widehat {\mathcal {G}} \to \mathcal {G}
.