  We give a summary of our recent studies of spatial and velocity biases of
galaxy-size halos in cosmological models. Recent progress in numerical
techniques made it possible to simulate halos in large volumes with a such
accuracy that halos survive in dense environments of groups and clusters of
galaxies. Dark matter halos in simulations look like real galaxies, and, thus,
can be used to study the biases - differences between galaxies and the dark
matter. For the standard LCDM model we find that the correlation function and
the power spectrum of galaxy-size halos at z=0 are antibiased on scales
r<5Mpc/h and k=(0.15-30)h/Mpc. The biases depend on scale, redshift, and
circular velocities of selected halos. Two processes seem to define the
evolution of the spatial bias: (1) statistical bias (or peak bias) and merger
bias (merging of galaxies, which happens preferentially in groups, reduces the
number of galaxies, but does not affect the clustering of the dark matter).
There are two kinds of velocity bias. The pair-wise velocity bias is
b_12=0.6-0.8 at r<5Mpc/h, z=0. This bias mostly reflects the spatial bias and
provides almost no information on the relative velocities of the galaxies and
the dark matter. One-point velocity bias is a better measure of the velocities.
Inside clusters the galaxies should move slightly faster (b_v=1.1-1.3) than the
dark matter. Qualitatively this result can be understood using the Jeans
equations of the stellar dynamics.
