  We establish the boundedness of solutions of reaction-diffusion systems with
quadratic (in fact slightly super-quadratic) reaction terms that satisfy a
natural entropy dissipation property, in any space dimension N>2. This bound
imply the smoothness of the solutions. This result extends the theory which was
restricted to the two-dimensional case. The proof heavily uses De Giorgi's
iteration scheme, which allows us to obtain local estimates. The arguments rely
on duality reasonings in order to obtain new estimates on the total mass of the
system, both in $L^{(N+1)/N}$ norm and in a suitable weak norm. The latter uses
$C^\alpha$ regularization properties for parabolic equations.
