<p>The fundamental study of ordered and disordered sphere packings provides deep physical insights about the structural and bulk physical properties of equilibrium and nonequilibrium phases of matter, and continues to offer tantalizing mathematical challenges. During the past three decades, there has been a flurry of developments on sphere packing problems across spatial dimensions in physics and mathematics. The purpose of this Review is to survey this progress in the case of frictionless identical (monodisperse) hard spheres, while placing it in the context of earlier classic investigations.</p>