In general, modal formalisms correspond to fragments of a full first-order (or sometimes higher-order) language over usual Kripke models. In this paper, the authors discuss which features of ‘modal fragments’ are responsible for these attractions. This paper presents a critical review of these problems, adding some new results about finite-variable fragments, including failures of the Łoś-Tarski preservation theorem. In particular, they prove decidability for a large ‘bounded fragment’ of predicate logic, and point out several applications.