Maxwell (1866) and Boltzmann (1872) developed a recipe to go from certain Newtonian laws of molecular dynamics to the Navier-Stokes system of gas dynamics. This recipe was controversial at the time. Mathematicians such as Hilbert, Klein, Poincare, and Zermelo were drawn into the debate. Hilbert featured it at the 1900 ICM in the articulation of his sixth problem, and made important contributions towards its resolution. The problem however remains largely open. Recent significant advances start with the DiPerna-Lions (1990) theory of global solutions to Boltzmann equations and lead to the Golse-Saint Raymond (2004) proof of the incompressible Navier-Stokes limit. This lecture will introduce the Boltzmann equation and survey some "new" connections to linear and weakly nonlinear gas dynamics that are the focus of recent research.