The paper is devoted to a study of generic representations (homomorphisms) of discrete countable groups
Γ
\Gamma
in Polish groups
G
G
, i.e., elements in the Polish space
R
e
p
(
Γ
,
G
)
\mathrm {Rep}(\Gamma ,G)
of all representations of
Γ
\Gamma
in
G
G
whose orbits under the conjugation action of
G
G
on
R
e
p
(
Γ
,
G
)
\mathrm {Rep}(\Gamma ,G)
are comeager. We investigate a closely related notion of finite approximability of actions on countable structures such as tournaments or
K
n
K_n
-free graphs, and we show its connections with Ribes–Zalesskii-like properties of the acting groups. We prove that
Z
\mathbb {Z}
has a generic representation in the automorphism group of the random tournament (i.e., there is a comeager conjugacy class in this group). We formulate a Ribes–Zalesskii-like condition on a group that guarantees finite approximability of its actions on tournaments. We also provide a simpler proof of a result of Glasner, Kitroser, and Melleray characterizing groups with a generic permutation representation.
We also investigate representations of infinite groups
Γ
\Gamma
in automorphism groups of metric structures such as the isometry group
Iso
(
U
)
\mbox {Iso}(\mathbb {U})
of the Urysohn space, isometry group
Iso
(
U
1
)
\mbox {Iso}(\mathbb {U}_1)
of the Urysohn sphere, or the linear isometry group
LIso
(
G
)
\mbox {LIso}(\mathbb {G})
of the Gurarii space. We show that the conjugation action of
Iso
(
U
)
\mbox {Iso}(\mathbb {U})
on
R
e
p
(
Γ
,
Iso
(
U
)
)
\mathrm {Rep}(\Gamma ,\mbox {Iso}(\mathbb {U}))
is generically turbulent, answering a question of Kechris and Rosendal.