The hard-core model from statistical physics is a randomly chosen independent set from a graph G (often the d-dimensional integer lattice), and the hard sphere model is a random sphere packing in Euclidean space.  I will describe a new method for bounding the partition function in each of these models based on analyzing the occupancy fraction: the expected fraction of vertices of G that appear in the random independent set or the expected density of the sphere packing.  Some of the applications of this method include new theorems in combinatorics and a surprising fact about spheres in dimension 24. I will also describe a possible approach to understanding the phase transition in hard-core systems via the occupancy fraction.