Context. Open clusters are convenient probes of the structure and history of the Galactic disk. They are also fundamental to stellar evolution studies. The second Gaia data release contains precise astrometry at the submilliarcsecond level and homogeneous photometry at the mmag level, that can be used to characterise a large number of clusters over the entire sky. Aims. In this study we aim to establish a list of members and derive mean parameters, in particular distances, for as many clusters as possible, making use of Gaia data alone. Methods. We compiled a list of thousands of known or putative clusters from the literature. We then applied an unsupervised membership assignment code, UPMASK, to the Gaia DR2 data contained within the fields of those clusters. Results. We obtained a list of members and cluster parameters for 1229 clusters. As expected, the youngest clusters are seen to be tightly distributed near the Galactic plane and to trace the spiral arms of the Milky Way, while older objects are more uniformly distributed, deviate further from the plane, and tend to be located at larger Galactocentric distances. Thanks to the quality of Gaia DR2 astrometry, the fully homogeneous parameters derived in this study are the most precise to date. Furthermore, we report on the serendipitous discovery of 60 new open clusters in the fields analysed during this study.