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The diffusion coefficient $D$ and the Ostwald solubility $\ensuremath{\alpha}$ have been measured for $^{133}\mathrm{Xe}$ atoms in water-sucrose solutions ranging in viscosity $\ensuremath{\eta}$ from 1 to 289 cP, i.e., 0.68% sucrose by weight. In water $D(20\ifmmode^\circ\else\textdegree\fi{}\mathrm{C})=(1.32\ifmmode\pm\else\textpm\fi{}0.05)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${\mathrm{cm}}^{2}$/sec, in agreement with the Stokes-Einstein (SE) law. Over the entire range studied the observed dependence is $D\ensuremath{\propto}{\ensuremath{\eta}}^{\ensuremath{-}0.63\ifmmode\pm\else\textpm\fi{}0.02}$, significantly departing from the SE law. In water $\ensuremath{\alpha}(20\ifmmode^\circ\else\textdegree\fi{}\mathrm{C})=0.108\ifmmode\pm\else\textpm\fi{}0.005$. For water-sucrose solutions $\ensuremath{\alpha}$ is approximately proportional to the volume fraction of water in solution.

The autocorrelation function of the density of a tagged particle is studied using the Mori formalism. The variables used are the collective conserved variables, the tagged-particle density, and bilinear products thereof. The case of point particles is considered in two dimensions, and, in three dimensions, self-diffusion by a particle of arbitrary size is treated. It is found that the bilinear-hydrodynamic approach automatically separates the self-diffusion coefficient of the tagged particle into a nonhydrodynamic part, and a hydrodynamic part which resembles the Stokes-Einstein law. In two dimensions, it is found that the mean-square displacement of a particle increases as $t\mathrm{ln}t$, and that certain natural redefinitions of the diffusion and friction coefficients leave Einstein's law invariant. In three dimensions, for a large particle, the Stokes-Einstein law is reproduced. The relation between the well-known ${t}^{(\frac{\ensuremath{-}3}{2})}$ "tails" on correlation functions, and the Stokes-Einstein law, is discussed.

The diffusion coefficient associated with the order-parameter fluctuations near the critical point of fluids is expected to vanish asymptotically as $D=\frac{\mathrm{RkT}}{6\ensuremath{\pi}\ensuremath{\eta}\ensuremath{\xi}}$, where $\ensuremath{\eta}$ is the viscosity, $\ensuremath{\xi}$ the correlation length, and $R$ a universal constant. Our experiments, using 3-methylpentane-nitroethane, yield $R=1.02\ifmmode\pm\else\textpm\fi{}0.06$, in agreement with the mode-coupling theory of critical fluctuations, but in disagreement with the value $\frac{R}{6\ensuremath{\pi}}=\frac{1.2}{6\ensuremath{\pi}}\ensuremath{\simeq}\frac{1}{5\ensuremath{\pi}}$, recently suggested by several investigators.