The Kostka numbers $K_{\lambda\mu}$ appear in combinatorics when expressing the Schur functions in terms of the monomial symmetric functions, as $K_{\lambda\mu}$ counts the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$. They also appear in representation theory as the multiplicities of weights in the irreducible representations of type $A$. Using a variety of tools from representation theory (Gelfand-Tsetlin diagrams), convex geometry (vector partition functions), symplectic geometry (Duistermaat-Heckman measure) and combinatorics (hyperplane arrangements), we show that the Kostka numbers are given by polynomials in the cells of a complex of cones. For fixed $\lambda$, the nonzero $K_{\lambda\mu}$ consist of the lattice points inside a permutahedron. By relating the complex of cones to a family of hyperplane arrangements, we provide an explanation for why the polynomials giving the Kostka numbers exhibit interesting factorization patterns in the boundary regions of the permutahedron. We will consider $A_2$ and $A_3$ (partitions with at most three and four parts) as running examples, with lots of pictures. I will also say a few words as to how some of the techniques used generalize to the case of Littlewood-Richardson coefficients. This is joint work with Sara Billey and Victor Guillemin.