In 2053 a manned mission to the moon, using seismic sensors, discovers that it is not a natural satellite*. The moon is actually an egg of Vespula ludicrousmegagiganto. Note that the name is not truly scientific - the shape of the creature is just vaguely wasp-like, and it stuck.
By 2071 and after extensive study, the scientific consensus is that - at some point between one and ten thousand years from now - the "egg" will hatch, and the creature will devour the earth to start its next phase of life. For obvious reasons we'd prefer this not to happen.
Is it possible to move the moon elsewhere using modern technology?
Since the science is imprecise, political will exists to start it moving ASAP, using all of the world's available resources. It could theoretically hatch at any time. They are looking to accomplish this with 2071 tech - consider this equivalent to what we have in 2015, perhaps with some optimizations but without any truly groundbreaking physics advancements. So no generated gravity or reactionless drives, for example.
Success is defined as "Put the moon into orbit around another planet within 500 years". Destroying it is officially considered too risky, although conspiracy theorists point out the tremendous scientific advances that could be gathered by studying the creature.
Ideally the movement would be accomplished without excessive impacts to the moon, but if that's not viable they will consider explosive drives.
*a competing theory is that it was a natural satellite, and the creature burrowed in and grew there over time. The difference is largely academic at this point.