  We report the results of a large scale computer simulation of a binary
supercooled Lennard-Jones liquid. We find that at low temperatures the curves
for the mean squared displacement of a tagged particle for different
temperatures fall onto a master curve when they are plotted versus rescaled
time $tD(T)$, where $D(T)$ is the diffusion constant. The time range for which
these curves follow the master curve is identified with the $\alpha$-relaxation
regime of mode-coupling theory (MCT). This master curve is fitted well by a
functional form suggested by MCT. In accordance with idealized MCT, $D(T)$
shows a power-law behavior at low temperatures. The critical temperature of
this power-law is the same for both types of particle and also the critical
exponents are very similar. However, contrary to a prediction of MCT, these
exponents are not equal to the ones determined previously for the divergence of
the relaxation times of the intermediate scattering function [Phys. Rev. Lett.
{\bf 73}, 1376 (1994)]. At low temperatures the van Hove correlation function
(self as well as distinct part) shows hardly any sign of relaxation in a time
interval that extends over about three decades in time. This time interval can
be interpreted as the $\beta$-relaxation regime of MCT. From the investigation
of these correlation functions we conclude that hopping processes are not
important on the time scale of the $\beta$-relaxation for this system and for
the temperature range investigated. We test whether the factorization property
predicted by MCT holds and find that this is indeed the case for all
correlation functions investigated. The distance dependence of the critical
amplitudes are in qualitative accordance with the ones predicted by MCT for
some other mixtures. The non-gaussian parameter for the self part of the van
