This article is an empirical study of the translatability of Polish jokes into English. The sample analysed consists of nine jokes, each of which belongs to one of the categories set up by Zabalbeascoa. The translatability of the jokes is discussed according to three approaches to translatability: Catford’s division between linguistic and cultural untranslatability, Ke’s sociosemiotic view, and Toury’s target-oriented approach. The conclusion is that each approach has shortcomings, notably so when applied in isolation. It is also concluded that much theoretical thinking on (un)translatability and, possibly, other features in Translation Studies are based on translation problems between language pairs that are known to the theorists.