It has long been speculated that a ninth planet (no, not Pluto) exists far beyond the orbit of Neptune. This so-called "Planet X" has been invoked to explain many phenomena. A Planet X was first suggested as a way to correct for perceived oddities in the orbits of Neptune and Uranus around the Sun, but it was later found that astronomers' expectations were incorrect due to the overestimation of the mass of Neptune. Planet X has also been proposed as an explanation for the irregular orbit of the trans-Neptunian object Sedna (a trans-Neptunian object is simply a body that orbits beyond Neptune). Sedna, which is likely a dwarf planet, has an orbit that takes it from 11 billion kilometers from the Sun at closest approach to 140 billion kilometers from the Sun at its farthest point. One theory for the origin of Sedna's irregular orbit is that it interacted with a large object (planet or star) in its past. A Planet X collision has also (unscientifically) been proposed as the cause of mass extinctions on Earth, and was suggested as a possible cataclysm to occur at the 2012 end of the Mayan calendar.
It appears that we can now rest easy, because the existence of a large planet in our solar system beyond Neptune has been largely disproven by NASA's WISE telescope (or the Wide-field Infrared Survey Explorer). WISE detects infrared light, which is lower energy than the visible light humans see, and is the kind of light that cooler astronomical objects tend to emit the most. Cool stars, brown dwarfs (or failed stars), and planets all emit lots of infrared light (as do humans!). WISE carried out two full scans of the sky in 2010 and 2011. By seeing how much an object moves in between the two images, we can see how far away it is (objects that are closer to us will appear to move more). And WISE failed to detect any Planet X. We can say definitively that there is no object the size of Saturn within 1.5 trillion kilometers of the Sun, and no Jupiter sized object within 4 trillion kilometers of the Sun. For reference, 1 trillion kilometers is about 7000 times the distance between the Earth and the Sun. If Planet X was there, WISE would have seen it.
WISE did, however, discover over 3,500 new stars and brown dwarfs within 500 light years of our Sun. That's basically our astronomical backyard! The observations used to make these discoveries were taken during WISE's 2010-2011 mission, after which WISE went into hibernation. It was recently reactivated and given a new mission and a new name: NEOWISE hunts for near-Earth objects (NEO), or comets and asteroids that could potentially pose a threat to humanity.