The interpretation of pulsar rotational glitches, the sudden increase in spin frequency of neutron stars, is a half-century-old challenge. The common view is that glitches are driven by the dynamics of the stellar interior, and connect in particular to the interactions between a large-scale neutron superfluid and the other stellar components. This thesis is corroborated by observational data of glitches and the post-glitch response seen in pulsars' rotation, which often involves very long timescales, from months to years. As such, glitch observables combined with consistent models incorporating the rich physics of neutron stars -- from the lattice structure of their crust to the equation of state for matter beyond nuclear densities -- can be very powerful at placing limits on, and reduce uncertainties of, the internal properties. This review summarises glitch observations, current data, and recent analyses, and connects them to the underlying mechanisms and microphysical parameters in the context of the most advanced theoretical glitch models to date.