Kostant’s partition function counts the number of ways to represent a particular weight (vector) as a nonnegative integral sum of positive roots of a Lie algebra (finite set of vectors). We provide two fundamental uses for this function. The first is associated with the computation of weight multiplicities in finite-dimensional irreducible representations of classical Lie algebras and the second is in the computation of volumes of flow polytopes. We present some recent results in both of these areas along with a connection to multiplex juggling.