  (Shortened) Weakly Interacting Massive Particles are often said to be the
best Dark Matter candidates. Studies have shown however that rather large Dark
Matter-photon or Dark Matter-baryon interactions could be allowed by cosmology.
Here we address the question of the role of the Dark Matter interactions in
more detail to determine at which extent Dark Matter has to be necessarily
weakly interacting. To this purpose, we compute the collisional damping (and
free-streaming) lengths of generic interacting Dark Matter candidates and
compare them to the scale of the smallest primordial structures known to exist
in the Universe. We obtain necessary conditions that any candidate must
satisfy. We point out the existence of new Dark Matter scenarios and exhibit
new damping regimes. For example, an interacting candidate may bear a similar
damping than that of collisionless Warm Dark Matter particles. The main
difference is due to the Dark Matter coupling to interacting (or even
freely-propagating) species. Our approach yields a general classification of
Dark Matter candidates which extends the definitions of the usual Cold, Warm
and Hot Dark Matter scenarios when interactions, weak or strong, are
considered.
