Explore the challenges and advancements in demonstrating quantum supremacy through BosonSampling in this comprehensive lecture by Scott Aaronson from the University of Texas at Austin. Delve into the comparison between BosonSampling and Random Circuit Sampling, examining their respective advantages and disadvantages. Gain insights into the recent developments, including the 2020 announcement from Hefei, China regarding a BosonSampling device with 50-70 detected photons. Investigate the theoretical open problems that have emerged as crucial in this field. Learn about key concepts such as unitary transformation, beam splitter networks, formal hardness results, and Gaussian Boson Sampling. Understand the issues surrounding photon losses, classical simulation, and verification methods like linear cross entropy. This in-depth discussion provides a thorough overview of BosonSampling's theory and practical applications in the quest for quantum supremacy.