  We investigate the sensitivity of the Laser Interferometer Space Antenna
(LISA) to the anisotropies of the Stochastic Gravitational Wave Background
(SGWB). We first discuss the main astrophysical and cosmological sources of
SGWB which are characterized by anisotropies in the GW energy density, and we
build a Signal-to-Noise estimator to quantify the sensitivity of LISA to
different multipoles. We then perform a Fisher matrix analysis of the prospects
of detectability of anisotropic features with LISA for individual multipoles,
focusing on a SGWB with a power-law frequency profile. We compute the noise
angular spectrum taking into account the specific scan strategy of the LISA
detector. We analyze the case of the kinematic dipole and quadrupole generated
by Doppler boosting an isotropic SGWB. We find that $\beta\, \Omega_{\rm
GW}\sim 2\times 10^{-11}$ is required to observe a dipolar signal with LISA.
The detector response to the quadrupole has a factor $\sim 10^3 \,\beta$
relative to that of the dipole. The characterization of the anisotropies, both
from a theoretical perspective and from a map-making point of view, allows us
to extract information that can be used to understand the origin of the SGWB,
and to discriminate among distinct superimposed SGWB sources.
