  We consider super-diffusive L\'evy walks in $d \geqslant 2$ dimensions when
the duration of a single step, i.e., a ballistic motion performed by a walker,
is governed by a power-law tailed distribution of infinite variance and finite
mean. We demonstrate that the probability density function (PDF) of the
coordinate of the random walker has two different scaling limits at large
times. One limit describes the bulk of the PDF. It is the $d-$dimensional
generalization of the one-dimensional L\'evy distribution and is the
counterpart of central limit theorem (CLT) for random walks with finite
dispersion. In contrast with the one-dimensional L\'evy distribution and the
CLT this distribution does not have universal shape. The PDF reflects
anisotropy of the single-step statistics however large the time is. The other
scaling limit, the so-called 'infinite density', describes the tail of the PDF
which determines second (dispersion) and higher moments of the PDF. This limit
repeats the angular structure of PDF of velocity in one step. Typical
realization of the walk consists of anomalous diffusive motion (described by
anisotropic $d-$dimensional L\'evy distribution) intermitted by long ballistic
flights (described by infinite density). The long flights are rare but due to
them the coordinate increases so much that their contribution determines the
dispersion. We illustrate the concept by considering two types of L\'evy walks,
with isotropic and anisotropic distributions of velocities. Furthermore, we
show that for isotropic but otherwise arbitrary velocity distribution the
$d-$dimensional process can be reduced to one-dimensional L\'evy walk.
