  The global existence of classical solutions to reaction-diffusion systems in
arbitrary space dimensions is studied. The nonlinearities are assumed to be
quasi-positive, to have (slightly super-) quadratic growth, and to possess a
mass control, which includes the important cases as mass conservation and mass
dissipation. Under these assumptions, the local classical solution is shown to
be global and, in case of mass conservation or mass dissipation, to have
$L^{\infty}$-norm growing at most polynomially in time. Applications include
skew-symmetric Lotka-Volterra systems and quadratic reversible chemical
reactions.
