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The dynamics of glass-forming liquids is heterogeneous and displays growing spatial correlations upon cooling. Whether such behavior arises from fluctuations in local structure or more complex forms of amorphous order is a highly debated question. To clarify this issue, we studied several model liquids within a coherent simulation framework based on the iso-configurational ensemble [1]. We found that the correlation between the preferred local structure and dynamic heterogeneity is system-dependent: it is pronounced in systems that deviate markedly from the mean-field picture of glassy dynamics and weak or absent in models that adhere to it to a good extent. I will review these results and then assess recent proposals to account for dynamic fluctuations using more genericmeasures of structure, such as overlap distributions and predictability analysis. Finally, I will characterize the structure of ultra-stable glassy samples of hard and soft spheres, which we recently equilibrated at large packing fractions using an optimized swap Monte Carlo algorithm [2]. [1] G. M. Hocky, D. Coslovich, A. Ikeda, D. R. Reichman, Phys. Rev. Lett. 113, 157801 (2014)[2] L. Berthier, D. Coslovich, A. Ninarello, M. Ozawa, arXiv:1511.06182 (2015)