In these proceedings we discuss advances in the theory and observation of magnetic fields in the intergalactic medium and in the cosmic web. We make the point that, despite perhaps unsurmountable obstacles in simulating a small-scale dynamo, currently most cosmological magnetohydrodynamical simulations paint a similar picture of magnetic field amplification in the cosmos. However, observations of magnetic fields in the intergalactic medium turn out to be very difficult. As a case in point, we present recent work on Faraday rotation measurement in the direction of a giant galaxy with the Low Frequency Array (LOFAR). These observations demonstrate the currently unique capability of LOFAR to measure Faraday rotation at the high accuracy and angular resolution required to investigate the magnetisation of large-scale structure filaments of the cosmic web.