The Boltzmann equation for dilute gas mixtures is formulated and the solution is expanded in powers of the gradients of the local density of each species, the temperature, the local velocity, and the local temperature of the gas. Thus hydrodynamic equations for mixtures of dilute gases are obtained with explicit expressions for the transport coefficients. These expressions can be expressed in forms that satisfy the Onsager reciprocal relations, and the rate of entropy production, to second order in the gradients, agrees with the predictions of non-equilibrium thermodynamics. The expressions for the transport coefficients, assuming various model interaction potentials, are compared with experimental results for binary mixtures.