Group operations on a fixed countably infinite universe, say ℕ, form a Polish space $\mathcal{G}$ (with the topology inherited from $\mathbb{N}^{\mathbb{N}\times\mathbb{N}}$). Thus we can view group properties as isomorphism-invariant subsets of $\mathcal G$, and it makes sense to ask: what properties are generic (in the sense of Baire category)? In my talk, I will address this question and if time permits, I may also say a few words about generic properties of compact groups.