  We study the biasing relation between dark-matter halos or galaxies and the
underlying mass distribution, using cosmological $N$-body simulations in which
galaxies are modelled via semi-analytic recipes. The nonlinear, stochastic
biasing is quantified in terms of the mean biasing function and the scatter
about it as a function of time, scale and object properties. The biasing of
galaxies and halos shows a general similarity and a characteristic shape, with
no galaxies in deep voids and a steep slope in moderately underdense regions.
At $\sim 8\hmpc$, the nonlinearity is typically $\lsim 10$ percent and the
stochasticity is a few tens of percent, corresponding to $\sim 30$ percent
variations in the cosmological parameter $\beta=\Omega^{0.6}/b$. Biasing
depends weakly on halo mass, galaxy luminosity, and scale. The time evolution
is rapid, with the mean biasing larger by a factor of a few at $z\sim 3$
compared to $z=0$, and with a minimum for the nonlinearity and stochasticity at
an intermediate redshift. Biasing today is a weak function of the cosmological
model, reflecting the weak dependence on the power-spectrum shape, but the time
evolution is more cosmology-dependent, relecting the effect of the growth rate.
We provide predictions for the relative biasing of galaxies of different type
and color, to be compared with upcoming large redshift surveys. Analytic models
in which the number of objects is conserved underestimate the evolution of
biasing, while models that explicitly account for merging provide a good
description of the biasing of halos and its evolution, suggesting that merging
is a crucial element in the evolution of biasing.
