We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group
G
G
over
R
\mathbb {R}
acting on an affine
R
\mathbb {R}
-variety
V
V
, we consider the induced dual action on the coordinate ring
R
[
V
]
\mathbb {R}[V]
and on the linear dual space of
R
[
V
]
\mathbb {R}[V]
. In this setting, given an invariant closed semialgebraic subset
K
K
of
V
(
R
)
V(\mathbb R)
, we study the problem of representation of invariant nonnegative polynomials on
K
K
by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on
R
[
V
]
\mathbb {R}[V]
by invariant measures supported on
K
K
. To this end, we analyse the relation between quadratic modules of
R
[
V
]
\mathbb {R}[V]
and associated quadratic modules of the (finitely generated) subring
R
[
V
]
G
\mathbb {R}[V]^G
of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional
K
K
-moment problem. Most of our results are specific to the case where the group
G
(
R
)
G(\mathbb {R})
is compact.