  The notion of `adhesion' has been advanced for the phenomenon of
stabilization of large-scale structure emerging from gravitational instability
of a cold medium. Recently, the physical origin of adhesion has been
identified: a systematic derivation of the equations of motion for the density
and the velocity fields leads naturally to the key equation of the `adhesion
approximation' - however, under a set of strongly simplifying assumptions. In
this work, we provide an evaluation of the current status of adhesive
gravitational clustering and a clear explanation of the assumptions involved.
Furthermore, we propose systematic generalizations with the aim to relax some
of the simplifying assumptions. We start from the general Newtonian evolution
equations for self-gravitating particles on an expanding Friedmann background
and recover the popular `dust model' (pressureless fluid), which breaks down
after the formation of density singularities; then we investigate, in a unified
framework, two other models which, under the restrictions referred to above,
lead to the `adhesion approximation'. We apply the Eulerian and Lagrangian
perturbative expansions to these new models and, finally, we discuss some
non-perturbative results that may serve as starting points for workable
approximations of non-linear structure formation in the multi-stream regime. In
particular, we propose a new approximation that includes, in limiting cases,
the standard `adhesion model' and the Eulerian as well as Lagrangian
first-order approximations.
