  We study the alignment of dark matter haloes with the cosmic web
characterized by the tidal and velocity shear fields. We focus on the alignment
of their shape, angular momentum and peculiar velocities. We use a cosmological
N-body simulation that allows to study dark matter halos spanning almost five
orders of magnitude in mass ($10^{9}$-$10^{14}$) $h^{-1}$$M_{\odot}$ and
spatial scales of $(0.5$-$1.0)$ $h^{-1}$ Mpc to define the cosmic web. We find
that the halo shape presents the strongest alignment along the smallest tidal
eigenvector, e.g. along filaments and walls, with a signal that gets stronger
as the halo mass increases. In the case of the velocity shear field only
massive halos $>10^{12}$ $h^{-1}$$M_{\odot}$ tend to have their shapes aligned
along the largest tidal eigenvector; that is, perpendicular to filaments and
walls. For the angular momentum we find alignment signals only for halos more
massive than $10^{12}$ $h^{-1}$$M_{\odot}$ both in the tidal and velocity shear
webs where the preferences are for it to be parallel to the middle eigenvector;
perpendicular to filaments and parallel to walls. Finally, the peculiar
velocities show a strong alignment along the smallest tidal eigenvector for all
halo masses; halos move along filaments and walls. In the velocity shear the
same alignment is present but weaker and only for haloes less massive than
$10^{12}$ $h^{-1}$$M_{\odot}$. Our results clearly show that the two different
algorithms we used to define the cosmic web describe different physical aspects
of non-linear collapse and should be used in a complementary way to understand
the effect of the cosmic web on galaxy evolution.
