In Fall 2019, a team at Google made the first-ever claim of "quantum computational supremacy"---that is, a clear quantum speedup over a classical computer for some task---using a 53-qubit programmable superconducting chip called Sycamore. Since then, a group at USTC in China has made several additional claims of quantum supremacy, using both superconducting qubits and "BosonSampling" (a proposal by me and Alex Arkhipov from 2011) with ~70 photons in an optical network. In addition to engineering, these experiments built on a decade of research in quantum complexity theory. This talk will discuss questions like: what exactly were the contrived computational problems that were solved? How does one verify the outputs using a classical computer? And crucially, how confident can we be that the problems are really classically hard? Scott J. Aaronson – ACM Prize in Computing 2020 The 8th Heidelberg Laureate Forum took place from September 20–23, 2021. #HLF21 The opinions expressed in the videos do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the videos.