Do Mormons believe heaven is outside of the universe or do they believe it's a city/kingdom floating in space or do they believe its on a planet in another star system?
Where do Mormons believe the current heaven is?
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Related to this is the question of, What do Mormons believe happens to us after we die?
Someday, like everyone else, your physical body will die. But your spirit does not die, it goes to the spirit world, where you will continue to learn and progress and may be with loved ones who have passed on.
Where is this "spirit world"? Wikipedia quotes Brigham Young:
It is right here...Do [spirits of the departed] go beyond the boundaries of the organized earth? No, they do not...Can you see it with your natural eyes? No. Can you see spirits in this room? No. Suppose the Lord should touch your eyes that you might see, could you then see the spirits? Yes, as plainly as you now see bodies.
So paradise is right here on earth but you can only access it with a spirit body. But notice I called it "paradise" and not heaven; the spirit world is not heaven. There are three heavens (or kingdoms) in Mormon cosmology: the celestial, terrestrial, and telestial. After the resurrection of the body, all people (except the very worst) will live in one of the three heavens. According to a vision of Joseph Smith, God dwells in the celestial kingdom, and Jesus and the Father are enthroned there.
As far as where the celestial kingdom is, the LDS church doesn't seem to have official teachings on it. The star/planet Kolob mentioned in the Book of Abraham is said to be "the heavenly body nearest to the throne of God," but there is much debate on the matter.
"Heaven" has three principal meanings.
Place of the righteous, pre-judgement
Jesus said to the thief on the cross, "To day shalt thou be with me in paradise" (Luke 23:43). Yet days later, when he appeared to Mary, he said "I am not yet ascended to my Father" (John 20:17). Mormons believe that the afterlife ("paradise") is distinct from the habitation of God.
Brigham Young taught that the spirit world is on Earth:
When you lay down this tabernacle, where are you going? Into the spiritual world....Where is the spirit world? It is right here....Can you see it with your natural eyes? No. Can you see spirits in this room? No. Suppose the Lord should touch your eyes that you might see, could you then see the spirits? Yes, as plainly as you now see bodies.
Discourses of Brigham Young, ed. Widtsoe
The separation between the physical and spiritual "dimensions" is frequently called the "veil."
Place of the righteous, post-judgement
Earth -- purified, and in the presence of God.
Blessed are the meek: for they shall inherit the earth.
Mormons call it the the celestial (heavenly) kingdom. Its inhabitants will dwell in the presence of God.
...the poor and the meek of the earth shall inherit it.
Therefore, it must needs be sanctified from all unrighteousness, that it may be prepared for the celestial glory;
For after it hath filled the measure of its creation, it shall be crowned with glory, even with the presence of God the Father;
Doctrine and Covenants 88:17-19
Throne of God
And I saw the stars, that they were very great, and that one of them was nearest unto the throne of God; and there were many great ones which were near unto it;
And the Lord said unto me: These are the governing ones; and the name of the great one is Kolob, because it is near unto me
P.S. If it seems strange that Mormons have astronomical references and unusual words like Kolob when referring to the location of heaven, it may be because you are used to having no specifics about its location. Think about it, (e.g. is heaven in the clouds?), and it might cause you to see the topic differently :)