This white paper is a summarising report of the Forum on monitoring the transient X-ray Universe in the multi-messenger era organized by the International Space Science Institute in Beijing (ISSI-BJ) on May 6-7, 2014. Time-domain astronomy will enter a golden era towards the end of this decade with the advent of major facilities across the electromagnetic spectrum and in the multi-messenger realms of gravitational wave and neutrino. In the soft X-ray regime, the novel micro-pore lobster-eye optics provides a promising technology to realise, for the first time, focusing X-ray optics for wide-angle monitors to achieve a good combination of sensitivity and wide field of view. In this context, Einstein Probe - a soft X-ray all-sky monitor - has been proposed and selected as a candidate mission of priority in the space science programme of the Chinese Academy of Sciences. This report summarises the most important science developments in this field towards 2020 and beyond and how to achieve them technologically, which were discussed at this brainstorming forum. It also introduces briefly the Einstein Probe mission, including its key science goals and mission definition, as well as some of the key technological issues.
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