  (Abridged) We develop the halo model of large-scale structure to include
triaxial dark matter haloes and their intrinsic alignments. As a direct
application we derive general expressions for the power spectrum. This we
calculate for two different triaxial density profiles. The first is a toy-model
that allows us to isolate the dependence of clustering on halo shape alone and
the second is the more realistic model of Jing & Suto (2002). In both cases, we
find that the effects of triaxiality are manifest as a suppression of power at
the level of ~5% on scales k~1-10 h/Mpc. When considered by mass, we find that
for the first model the power is suppressed and that this is more significant
for high mass haloes. For the Jing & Suto model, we find a similar suppression
on large scales, but followed by a sharp amplification on small scales at the
level of ~10-15%, for the highest mass haloes.
  One of the important features of our formalism is that it allows for the
self-consistent inclusion of the intrinsic alignments of haloes. The alignments
are specified through the correlation function of halo seeds. We develop a
useful toy model for this and make estimates of the alignment contribution to
the power spectrum. Further, through consideration of the case where all haloes
are perfectly aligned, we calculate the maximum contribution to the clustering
and find the hard limit of <10%.
