  We review some exact results for the motion of a tagged particle in simple
models. Then, we study the density dependence of the self diffusion
coefficient, $D_N(\rho)$, in lattice systems with simple symmetric exclusion in
which the particles can jump, with equal rates, to a set of $N$ neighboring
sites. We obtain positive upper and lower bounds on
$F_N(\rho)=N((1-\r)-[D_N(\rho)/D_N(0)])/(\rho(1-\rho))$ for $\rho\in [0,1]$.
Computer simulations for the square, triangular and one dimensional lattice
suggest that $F_N$ becomes effectively independent of $N$ for $N\ge 20$.
