In the past decade we have started to explore extragalactic and intergalactic space using millisecond-duration radio flashes called `fast radio bursts’ (FRBs). These cosmological signals are surprisingly abundant: there is likely an FRB occurring somewhere on the sky at least once every minute. But what is producing them? Thanks to a new generation of wide-field radio telescopes, several FRBs per day are now being discovered. Novel high-time-resolution observations using radio interferometers are now also pinpointing FRB locations and providing host galaxy associations. More than a decade since the famous `Lorimer burst’, we are now making rapid progress in our understanding of the FRB phenomenon. In this talk, I will focus on how observations with Arecibo, the European VLBI Network (EVN) and the Low-Frequency Array (LOFAR) are shedding light on the nature of FRB sources. These observations have provided milli-arcsecond localisations and microsecond-resolution polarimetry to decode the source model and emission mechanism.