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.