  We present distribution functions and spatial correlations of the shapes of
dark matter halos derived from Hubble Volume simulations of a LambdaCDM
universe. We measure both position and velocity shapes within spheres
encompassing mean density 200 times the critical value, and calibrate small-N
systematic errors using Poisson realizations of isothermal spheres and higher
resolution simulations. For halos more massive than 3x10^{14} Msun/h, the shape
distribution function peaks at (minor/major, intermediate/major) axial ratios
of (0.64,0.76) in position, and is rounder in velocity, peaking at (0.72,0.82).
Halo shapes are rounder at lower mass and/or redshift; the mean minor axis
ratio in position follows <c/a>(M,z) = c_{15,0} [1-\alpha\ln(M/10^{15}Msun/h)]
(1+z)^{-\epsilon}, with c_{15,0}=0.631 \pm 0.001, \alpha=0.023 \pm 0.002 and
\epsilon=0.086 \pm 0.004. Position and velocity principal axes are well aligned
in direction, with median alignment angle $22^\circ$, and the axial ratios in
these spaces are correlated in magnitude. We investigate mark correlations of
halo pair orientations using two measures: a simple scalar product shows $\ge
1%$ alignment extending to $30 \hinv \mpc$ while a filamentary statistic
exhibits non-random alignment extending to scales $\sims 200 \hinv \mpc$, ten
times the sample two-point correlation length and well into the regime of
negative two-point correlation. Cluster shapes are unaffected by the
large-scale environment; the shape distribution of supercluster members is
consistent with that of the general population.
