We use causality to derive a number of simple and universal constraints on dispersion relations, which describe the location of singularities of retarded two-point functions in relativistic quantum field theories. We prove that all causal dissipative dispersion relations have a finite radius of convergence in cases where stochastic fluctuations are negligible. We then give two-sided bounds on all transport coefficients in units of this radius, including an upper bound on diffusivity.