The second Gaia Data Release (DR2) contains astrometric and photometric data for more than 1.6 billion objects with mean Gaia G magnitude <20.7, including many Young Stellar Objects (YSOs) in different evolutionary stages. In order to explore the YSO population of the Milky Way, we combined the Gaia DR2 data base with Wide-field Infrared Survey Explorer (WISE) and Planck measurements and made an all-sky probabilistic catalogue of YSOs using machine learning techniques, such as Support Vector Machines, Random Forests, or Neural Networks. Our input catalogue contains 103 million objects from the DR2xAllWISE cross-match table. We classified each object into four main classes: YSOs, extragalactic objects, main-sequence stars, and evolved stars. At a 90 per cent probability threshold, we identified 1 129 295 YSO candidates. To demonstrate the quality and potential of our YSO catalogue, here we present two applications of it. (1) We explore the 3D structure of the Orion A star-forming complex and show that the spatial distribution of the YSOs classified by our procedure is in agreement with recent results from the literature. (2) We use our catalogue to classify published Gaia Science Alerts. As Gaia measures the sources at multiple epochs, it can efficiently discover transient events, including sudden brightness changes of YSOs caused by dynamic processes of their circumstellar disc. However, in many cases the physical nature of the published alert sources are not known. A cross-check with our new catalogue shows that about 30 per cent more of the published Gaia alerts can most likely be attributed to YSO activity. The catalogue can be also useful to identify YSOs among future Gaia alerts.