  Using six high resolution dissipationless simulations with a varying box size
in a flat LCDM universe, we study the mass and redshift dependence of dark
matter halo shapes for M_vir = 9.0e11 - 2.0e14, over the redshift range z=0-3,
and for two values of sigma_8=0.75 and 0.9. Remarkably, we find that the
redshift, mass, and sigma_8 dependence of the mean smallest-to-largest axis
ratio of halos is well described by the simple power-law relation <s> = (0.54
+- 0.02)(M_vir/M_*)^(-0.050 +- 0.003), where s is measured at 0.3 R_vir and the
z and sigma_8 dependences are governed by the characteristic nonlinear mass,
M_*=M_*(z,sigma_8). We find that the scatter about the mean s is well described
by a Gaussian with sigma ~ 0.1, for all masses and redshifts. We compare our
results to a variety of previous works on halo shapes and find that reported
differences between studies are primarily explained by differences in their
methodologies. We address the evolutionary aspects of individual halo shapes by
following the shapes of the halos through ~100 snapshots in time. We determine
the formation scalefactor a_c as defined by Wechsler et al. (2002) and find
that it can be related to the halo shape at z = 0 and its evolution over time.
