Quite a large number of people accept, either fully or partially, what is called reincarnation. But for many there are also some confusion or misunderstanding.
This talk attempts to shed some light on the subject. Some verses which I quoted this subject:
As the embodied soul continuously passes, in this body, from boyhood to youth to old age, the soul similarly passes into another body at death. A sober person is not bewildered by such a change. Bhagavad-gita 2.13
For the soul there is neither birth nor death at any time. He has not come into being, does not come into being, and will not come into being. He is unborn, eternal, ever-existing and primeval. He is not slain when the body is slain. Bhagavad-gita 2.20
As a person puts on new garments, giving up old ones, the soul similarly accepts new material bodies, giving up the old and useless ones. Bhagavad-gita 2.22
The living entity in the material world carries his different conceptions of life from one body to another as the air carries aromas. Thus he takes one kind of body and again quits it to take another. Bhagavad-gita 15.8
The caterpillar transports itself from one leaf to another by capturing one leaf before giving up the other. Similarly, according to his previous work, the living entity must capture another body before giving up the one he has. This is because the mind is the reservoir of all kinds of desires. Srimad Bhagavatam 4.29.76-77