Entanglement is what makes many-body systems special. It is the key aspect of their enormous informational content, which implies a seemingly insurmountable hardness in simulating them; but also a whole new world of possibilities for storing and transforming information with them. In this talk I will discuss how multipartite entanglement -- i.e. the most collective form of entanglement -- can be produced by the non-equilibrium dynamics of lattice spin models, realized experimentally in several platforms of quantum simulation. I shall discuss the role of connectivity of the interactions among spins (i.e. the range of interactions, and the number of dimensions); and of the nature of elementary excitations, which fundamentally establish entanglement with their dynamics. In turn, I will discuss how the spectrum of the elementary excitations can be read from the time-dependent dynamics, suggesting a new form of spectroscopy for synthetic quantum matter. Our theoretical results will be presented in parallel with experimental ones obtained in Rydberg-atom arrays, representing one of the most promising platforms to explore many-body entanglement.