Explore the fascinating world of sphere packing problems in this comprehensive lecture by Abhinav Kumar at the International Centre for Theoretical Sciences. Delve into the centuries-old geometric challenge of arranging spheres in space to maximize density, and discover its connections to number theory, communication theory, and physics. Learn about known solutions in low dimensions, including the hexagonal planar packing and the Hales theorem. Investigate higher-dimensional packings, non-lattice arrangements, and the intriguing E8 and Leech lattices. Examine upper and lower bounds, lattice theory, and asymptotic behavior in high dimensions. Explore related concepts like spherical codes, the kissing number problem, and energy minimization. Gain insights into open problems and recent advancements in this active area of mathematical research.