  We carry on a systematic study of nearly Sasakian manifolds. We prove that
any nearly Sasakian manifold admits two types of integrable distributions with
totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional
nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a
contact manifold. Focusing on the $5$-dimensional case, we prove that there
exists a one-to-one correspondence between nearly Sasakian structures and a
special class of nearly hypo $SU(2)$-structures. By deforming such a
$SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Further we
prove that both nearly Sasakian and Sasaki-Einstein $5$-manifolds are endowed
with supplementary nearly cosymplectic structures. We show that there is a
one-to-one correspondence between nearly cosymplectic structures and a special
class of hypo $SU(2)$-structures which is again strictly related to
Sasaki-Einstein structures. Furthermore, we study the orientable hypersurfaces
of a nearly K\"{a}hler 6-manifold and, in the last part of the paper, we define
canonical connections for nearly Sasakian manifolds, which play a role similar
to the Gray connection in the context of nearly K\"{a}hler geometry. In
dimension $5$ we determine a connection which parallelizes all the nearly
Sasakian $SU(2)$-structure as well as the torsion tensor field. An analogous
result holds also for Sasaki-Einstein structures.
