The way I understand Buddhism of the Old School (Theravada) is that there is no "self" or "soul". We consist of "collections of aggregates". On dying, these collections are dispersed and are reassembled in other collections, having no direct relationship with the former collection. It is like when you light a candle with another candle. The newly lit candle does not arise from the old one, but is another candle. For discussions, however, it is convenient to speak of "self" and "I" etc.
What kind of beings that emerge on this reassembling is determined by our present lives. Every deliberate action (karma) bears fruits, bad fruit from bad actions and good from good. From good fruit emerges a being in a better situation, while bad fruit may cause an animal. The new individual is the result of previous actions, not a "soul" taking on a new existence.
In spite of this, I think that most Buddhists believe that the new-born being in some way is identical to the old one and a continuation of the old one.
This has puzzled me for a long time. If there is no relation between "me" and the resulting being, why should I bother about my actions? I suppose that the answer is, that we out of compassion should avoid causing beings with miserable lives to emerge.