A new,

Context. Blue straggler stars are exotic objects present in all stellar environments whose nature and formation channels are still partially unclear. They seem to be particularly abundant in open clusters (OCs), thus offering a unique chance to tackle these problems statistically. Aims. We aim to build up a new and homogeneous catalogue of blue straggler stars (BSS) in Galactic OCs using Gaia to provide a more solid assessment of the membership of these stars. We also aim to explore possible relationships of the straggler abundance with the parent cluster’s structural and dynamical parameters. As a by-product, we also search for possible yellow straggler stars (YSS), which are believed to be stragglers in a more advanced evolution stage. Methods. We employed photometry, proper motions, and parallaxes extracted from Gaia DR2 for 408 Galactic star clusters and searched for stragglers within them after performing a careful membership analysis. Results. The number of BBS emerging from our more stringent, selection criteria turns out to be significantly smaller than in previous versions of this catalogue. OCs are therefore not a preferable environment for these kinds of stars anymore. In addition, we found that BSS start to appear in clusters with ages larger than log(t) ∼ 8.7 and are therefore absent in very young star clusters. Conclusions. The present catalogue supersedes the previous ones in several ways: membership assessment, number of stragglers found, and so forth. The new list includes 897 BSS and 77 YSS candidates in 408 OCs. We expect this catalogue to be the basis for a new round of studies of BSS and YSS