Itinerant bosonic systems are of great interest but hard to tackle due to the large number of modes. We will use cutting-edge methods in theory and experiment to advance the understanding of these systems. On the one hand, we will advance matter-wave microscopy to image continuous quantum gases with sub-micron effective resolution, enabling us to probe local information spreading. On the other hand, we plan to develop new NQS-based techniques for numerical simulations of bosonic systems in the challenging realms of two spatial dimensions and continuous space.