  A theory of clustering of inertial particles advected by a turbulent velocity
field caused by an instability of their spatial distribution is suggested. The
reason for the clustering instability is a combined effect of the particles
inertia and a finite correlation time of the velocity field. The crucial
parameter for the clustering instability is a size of the particles. The
critical size is estimated for a strong clustering (with a finite fraction of
particles in clusters) associated with the growth of the mean absolute value of
the particles number density and for a weak clustering associated with the
growth of the second and higher moments. A new concept of compressibility of
the turbulent diffusion tensor caused by a finite correlation time of an
incompressible velocity field is introduced. In this model of the velocity
field, the field of Lagrangian trajectories is not divergence-free. A mechanism
of saturation of the clustering instability associated with the particles
collisions in the clusters is suggested. Applications of the analyzed effects
to the dynamics of droplets in the turbulent atmosphere are discussed. An
estimated nonlinear level of the saturation of the droplets number density in
clouds exceeds by the orders of magnitude their mean number density. The
critical size of cloud droplets required for clusters formation is more than
$20 \mu$m.
