In class 10 when we studied magnetism in physics for the first time we were told that magnets exist in the form of dipoles only, that is, magnets have a north and south pole. If you break a magnet into two pieces, these two pieces behave as individual magnets. If you break divide the two pieces and keep on dividing till you reach the molecular level, you will find that each piece behaves as an individual magnet. We were also told that magnetic monopoles, ie magnets with either a north or a south pole, do not exist at all. But there isn't any fundamental reason that does not allow the existence of magnetic monopoles in our universe. It's just that we have not observed such particles in our universe at all. Even James Clerk Maxwell, the man who is known for uniting electricity and magnetism under one set of equations (known as Maxwell's equations of electromagnetism), could not prove the existence of monopoles through the use of his iconic equations. So why can't we observe these particles?
To hunt for these particles we need to go back to the Big Bang. The Planck era of the universe was characterised by extremely high temperatures and density. In these times, the laws of physics were radically different from the ones we have now. Due to the high temperature and density, the fundamental forces of nature were once a single force. If we travel back in time we will arrive at a point where the energy and temperature was high enough for the electromagnetic and weak nuclear forces to be one force known as the electroweak force. If we go back further where the energy and temperature was even higher, we will see the electroweak and the strong nuclear force merge to become one force. This theory of the merger of the strong, weak and electromagnetic forces is known as the Grand Unified Theory.
The splitting of the three GUT forces is theorised to be a violent, and highly energetic phenomenon that sent "space quakes" throughout the universe that left small defects in the fabric of space time. Imagine cracks and fissures on buildings caused due to an earthquake, that have endured for a long time. Just like how earthquakes are usually spontaneous phenomena, the splitting of GUT forces in the early universe was a spontaneous and violent phenomena. The small defects that were left behind were diverse but one such defect conformed to the properties and characteristics of magnetic monopoles. Since the split covered the entire universe such defects should have spread out in large numbers and thus we should have had magnetic monopoles all around the universe. But ironically, this isn't true and we do not observe any magnetic monopoles in the universe.
This problem caused a lot of worry among the cosmologists around the world when it cropped up in the 1960s. Cosmologist, Alan Guth proposed a solution to this problem of magnetic monopoles. He proposed that the splitting of the GUT forces which could've caused the monopoles to plop into existence could also have caused them to become unobservable. This could've been possible due a rapid and massive expansion of the universe caused by the energy released by GUT force split. The inflation of the universe made it exponentially larger than what it was before inflation began. This means that the universe is actually much, much larger than what we can observe; larger by a factor of 10^62. Therefore, it does not matter how many monopoles were created at the beginning of the universe, the size of the universe is extremely large compared to that number and therefore when observed from Earth, we can only observe one magnetic monopole anywhere, a number which is too low to observe properly and a probable reason why we have still not observed monopoles.
However, this solution is purely hypothetical. Even the problem is pure conjecture at this point and has no experimental verification of any kind. However, if magnetic monopoles do exist then that would revolutionise the field of physics. The proof of existence of magnetic monopoles is a proof of validity of string theory, which theorises that all particles in the universe exist in the form of one dimensional strings and not dimensionless units. But, as stated earlier, the presence of magnetic monopoles is still a subject of intense research and we can only wait and watch for any results.