Short answer: Yes, rebirth does happen and as humans, it is very important for us to know about it and learn to get out of it.
Long answer: Most of the major details (about the soul, proofs of soul, the science behind rebirth, way to get out of the rebirth cycle, etc.) have been provided below. Read till the end to get the most out of this answer.
The most reliable and authentic source to gain knowledge about reincarnation or rebirth is Bhagavad Gita. It gives a clear, logical explanation of the science of reincarnation. Lord Krishna spoke Bhagavad Gita to Arjuna on a battlefield – which is a suitable place to discuss reincarnation as people confront death face to face in combats and common questions about life and the afterlife may arise. Before jumping into the fascinating details of reincarnation, it is important to understand the eternal part of every living being, i.e., the soul.
Soul – the Eternal Living Part
Lord Krishna in Bhagavad Gita (chapter 2) says, “That which pervades the entire body you should know to be indestructible. No one is able to destroy that imperishable soul… The soul is unbreakable and incombustible; it can neither be dampened nor dried. It is everlasting, in all places, unalterable, immutable, and primordial… Some look on the soul as amazing, some describe him as amazing, and some hear of him as amazing, while others, even after hearing about him, cannot understand him at all.”
Proofs of the Presence of the Soul
Bhagavad Gita says that the soul is immortal and is never destroyed. As the sun alone illuminates all this universe, so does the soul within the body illuminate the entire body by consciousness. Without this consciousness, the body is nothing but a lump of dead matter. It is said that the soul is the driver of the body. When it leaves the body, all its functions stop, and it becomes useless within a fraction of a second. This can be observed when someone dies.
Also, as we can see with time, from childhood to adulthood, the body keeps changing, but something remains the same. The distinction between the conscious self (soul) and the body becomes more visible when we grow older. If we think more deeply about this, this can act as another proof of the presence of a living soul within us.
Many scientists, researchers, and famous people have complete faith in the presence of the soul. As there are a lot of proofs and references but less space to write so this portion will be covered in another answer.
Core Principles behind Reincarnation
Bhagavad Gita says that souls cannot be slain even if the body is slain. But if this is the case, then what exactly happens to the soul? Lord Krishna beautifully explains, "The soul is neither born, nor does it ever die, nor having once existed, does it ever cease to be. The soul is without birth, eternal, immortal, and ageless. It is not destroyed when the body is destroyed… Just as the embodied soul continuously passes from childhood to youth to old age, similarly, at the time of death, the soul passes into another body. The wise are not deluded by this… As a person sheds worn-out garments and wears new ones, likewise, at the time of death, the soul casts off its worn-out body and enters a new one." These are very important lines that form the basic principles behind reincarnation. So, we can say that reincarnation is the transmigration of the soul from one body to another.
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This is a great answer, however, a couple of important notes.
Note 1
Not only the physical body of a human is constantly changing, but also the sense of consciousness or the quality of a soul.
The quality of the soul fully depends on the energetic spectrum of the soul as well as the quantity of these energies.
This follows from the Universal Laws of Change and Evolution.
The quality of soul energies can even change to the opposite when a person makes such a Spiritual Choice, which changes the former Spiritual Direction* to the opposite.
For example, a person who follows the Values of Light makes a Spiritual Choice towards Darkness.
Therefore, we can call the Spiritual Choice also a Spiritual Trial.
VALUES OF DARKNESS
SLAVERY
HATRED
WAR
COWARDICE
FALSE
INJUSTICE
DESPAIR
CRUELTY
CHAOS
GREED
RIVALRY
VALUES OF LIGHT
FREEDOM
LOVE
PEACE
COURAGE
TRUTH
JUSTICE
HOPE
HUMANITY
HARMONY
GENEROSITY
FRATERNITY
A good example of how a highly Spiritual person had transformed into a vampire could be found in the movie "Bram Stoker's Dracula" by Francis Ford Coppola. Please find details in my review of the movie.
Note 2
According to the newest information, a soul can be destroyed.
However, this can be done only by the decision of the Supreme Creation, i.e. Creator of this soul.
This severe punishment is applied to parasitic people who do not want to give anything to others, but simply consume the resources of society or family and do this for four human lives in a row.
Other souls who do not deserve further existence are unnecessarily cruel souls . They also get a number of chances to improve themselves. If these chances are ignored for several lifetimes, the souls receive the death penalty.
During the procedure all soul's energies are cleared from their matrix and the personality leaves existence. This procedure is extremely painful.
Note 3
In my feature screenplay The Return of Melissa the protagonist is a scientist who didn't understand the spiritual matters regarding a soul, although he believed in God, and even competed with him.
With his ignorance regarding soul he went to an experiment, that not only horrified his psyche but also proved him the existence of a soul and the power of its Creator.
Find more about the destiny of the scientist in the screenplay.
* Spiritual Direction.
There are two Spiritual Directions: towards Light and towards Darkness. Between them is sub-direction: Degradation.