The primary thesis of this paper is that, contrary to popular views, the translation of proper names is a non-trivial question, closely related to the problem of the meaning of the proper name. It aims to show what happens to proper names in the process of translation, particularly from English into Hungarian, to systematise and, within the frames of relevance theory, to explain the phenomena in question. It is suggested that in translating a proper name translators have four basic operations at their disposal: transference, translation proper, substitution and modification, which are defined here and explained in relevance-theoretic terms. The paper presents two case studies, which attempt to explain the treatment of proper names in the Hungarian translations of Kurt Vonnegut's Slaughterhouse-Five and J. F. Cooper's The Last of the Mohicans. The analysis is based on the assumption that translation is a special form of communication, aimed at establishing interpretive resemblance between the source text and the target text, governed by the principle of optimal resemblance (Sperber and Wilson 1986; Gutt 1991). The findings seem to confirm the claim that proper names behave in a largely predictable way in translation: the particular operations chosen to deal with them are a function, partly, of the semantic content they are loaded with in the source context and, partly, of considerations of how this content may be preserved in the target communication situation, including elements like the specific audience, intertextual relationships and translation norms, in consistency with the principle of relevance.