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An interest in some property that is currently assigned to a known person. If person A makes a Will leaving certain propery to whichever of my children marries first, then until they marry, A's children have no vested interest. When one of them marries, the property is 'vested in' that person. Until it vests, the children have only acontingent interest, that is, the possibility of gaining some interest.
A vested interest need not be in possession; in other words, a person can have an interest which becomes possessory at some point in the future. If, for example, A leaves property to my wife for life, then to my daughter D, D has a vested interest 'in remainder'. It is a real interest, and will certainty come into D's possession in the normal course of events. D could even, in principle, raise a Mortgage on that interest, if anyone were willing to lend on such security.
If, on the other hand, A's will left the property to my wife for life, then to which ever of my children marries first, if none of the children are married when A dies, none of them have a vested interest. They all have, at best, a contingent interest, which is of no particular value. As soon as one of them marries, the interest is vested in that person, and all the other contingent interests are destroyed. If the wife is still alive at that point, then the interest which vests is an interest in possession; if not, it is an interest in remainder, which will become possessory in due course.
In reality, no contingent interest is recognized as a separate species of legal right, and these interests must be given effect by a Trust. This means that the usual rules of trust creation and administration apply. In particular, the RuleAgainstPerpetuities will come into effect if the contingent interests do not vest in somebody within a particular time.