A theory of non-Markovian translational Brownian motion in a Maxwell fluid is developed. A universal kinetic equation for the joint probability distribution of position, velocity, and acceleration of a Brownian particle is derived directly from the extended dynamic equations for the system. Unlike the extended Fokker-Planck equation which corresponds to Mori-Kubo generalized Langevin equation and provides only with calculations of one-time moments, the universal kinetic equation obtained gives complete statistical description of the process. In particular, an exact generalized Fokker-Planck equation in the velocity space valid for any time instant is derived for the free non-Markovian Brownian motion. It shows that both the "master telegraph" and the respective kinetic equations, obtained in the molecular theory of Brownian motion, are type of approximations. The long and short time behavior of velocity and force correlations for a free Brownian particle is investigated in the general case of a nonequilibrium initial value problem. A corresponding diffusion equation in the coordinate space, and the generalized Einstein relation between the diffusion coefficient and the mobility are derived.