We develop closed-form transition density approximations for multivariate affine jump diffusion processes using polynomial expansion techniques. The approximations converge in L2 for a fixed time horizon, provided that the processes with support on R+ satisfy non-attainment conditions. Empirical applications in portfolio credit risk, likelihood inference, and option pricing using the (integrated) square-root jump diffusion, and Heston's model indicate that the approximations perform very accurately. The expansions are extremely fast to evaluate and numerically stable compared to Fourier inversion.