Experiments with ultracold atoms enable the preparation of strongly interacting quantum matter with widely tunable parameters. Exploiting the control over particle statistics, interactions and dimensionality, we can now design synthetic quantum systems and study them under extreme conditions. The most non-classical feature of strongly correlated phases is the presence of entanglement between the constituent particles. I will discuss novel methods that make use of single-particle control and atom-resolved readout in ultracold quantum gases to directly measure the presence of entanglement in bosonic and fermionic systems. The latter is enabled by a technique to detect single atoms in free space and to characterize few-body systems through their spin-resolved momentum correlation functions. We find that strong correlations in conjugate bases verify the presence of entanglement. Applying our tools to mesoscopic quantum systems can give insight into the nature of unconventional superconductivity and the properties of topological quantum matter.