Things like our body, mind, soul, spirit/consciousness, held together by the desire for "self". As long as that desire remains, the illusion of "self" (as expressed through rebirth of the body, mind, etc.) continues the unending process of rebirth.
By what mechanism can these things continue on after death?
I personally believe in a non-permanent soul that exists through many incarnations. This is actually a subtle body on the Causal realm of nature that causes the new incarnations. Perhaps the western concept of a permanent soul is getting in the way here.