The difficulty of tackling the out-of-equilibrium dynamics of gauge theories on classical computers is spurring a worldwide effort to solve them on dedicated quantum simulator devices. In this talk, I will discuss recent progress towards quantum simulation of gauge theories. First, I will present recent experiments in cold atoms [1,2], one of which has realized a many-body gauge theory in a 71-site Hubbard model and has certified the fulfilment of Gauss’s law for the first time . Moreover, I will discuss our ongoing theoretical effort to quantify and mitigate the influence of microscopic violations of the local gauge symmetry [3–5]. Through these discussions, I will aim at outlining a roadmap towards mature and practically relevant quantum simulation of gauge theories.
 A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, F. Jendrzejewski, Realizing a scalable building block of a U(1) gauge theory with cold atomic mixtures, Science 367, 1128-1130 (2020).
 Bing Yang, Hui Sun, Robert Ott, Han-Yi Wang, Torsten V. Zache, Jad C. Halimeh, Zhen-Sheng Yuan, Philipp Hauke, Jian-Wei Pan, Observation of gauge invariance in a 71-site quantum simulator, Accepted in Nature, arXiv:2003.08945 [cond-mat.quant-gas] (2020).
 J. C. Halimeh, P. Hauke, Reliability of lattice gauge theories, Phys. Rev. Lett.125, 030503 (2020), Staircase prethermalization and constrained dynamics in lattice gauge theories, arXiv:2004.07248 [cond-mat.quant-gas] (2020), Origin of staircase prethermalization in lattice gauge theories, arXiv:2004.07254 [cond-mat.str-el] (2020).
 J. C. Halimeh, R. Ott, I. P. McCulloch, B. Yang, P. Hauke, Robustness of gauge-invariant dynamics against defects in ultracold-atom gauge theories, Accepted in Phys. Rev. Research arXiv:2005.10249 [cond-mat.quant-gas] (2020).