  We investigate the connection between local structure and dynamical
heterogeneity in supercooled liquids. Through the study of four different
models we show that the correlation between a particle's mobility and the
degree of local order in nearby regions is highly system dependent. Our results
suggest that the correlation between local structure and dynamics is weak or
absent in systems that conform well to the mean-field picture of glassy
dynamics and strong in those that deviate from this paradigm. Finally, we
investigate the role of order-agnostic point-to-set correlations and reveal
that they provide similar information content to local structure measures, at
least in the system where local order is most pronounced.
