Supersymmetry assigns every particle a supersymmetric partner, whose spin differs by half.  The photon (spin "1"), for example, has a supersymmetric partner the photino (spin "½").  Other particles follow the same pattern; bosons (integer spin) have fermionic (fractional spin) supersymmetric partners, and fermions have bosonic supersymmetric partners.  Unfortunately, none has ever been found.   If supersymmetry exists, it must be a broken symmetry to permit superpartner particles that are heavier than corresponding Standard Model partners.  Thus, it is likely that many of the supersymmetric partners are so heavy that the particle colliders presently in use are not powerful enough to produce them.  This "broken" symmetry retains the benefits of supersymmetry, allowing the standard model to describe particle physics down to the Planck scale.  Supersymmetry is a necessary ingredient for most versions of String Theory, but does not itself require a String Theory.

Physics

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Quantum Gravity

When quantum mechanics and special relativity were being combined, one of the results was a whole new raft of particles called anti-particles; one for every particle known.  Essentially, this doubled the number of known particles.  Subsequently, as physicists discovered new particles, their anti-particle cousins showed up too.  Now, in the attempt to combine quantum physics with general relativity, and address the issue of gravity on the quantum scale, the need to double the number of particles again looms large!  This is the theory of Supersymmetry.  A version of supersymmetry was suggested back in 1966, but was not pursued until 1971 through 1974 to become what we see today.  Its first application in the standard model was proposed in 1981, and is called the Minimal Supersymmetric Standard Model.  But what does symmetry mean in this context?  Consider a sphere.  However it is moved, it looks the same; it is symmetrical.  A cube, however, looks different as it is, for example, rotated, so is not symmetrical.  In physics, things are a little more complex.  General relativity states that all physical laws are the same in all frames of reference, even those in motion and under acceleration.  This is a symmetry.
The spin-0 superpartners of fermions are called sfermions.  There are two classes.  The quark superpartners are called squarks.  Thus, for example, an "up quark" has an "up squark" as its superpartner, and a "strange anti-quark" would have a "strange anti-squark".  Lepton  superpartners are called sleptons, and their superpartners are called, for example, the selectron, which is superpartner of an electron, the smuon, which is superpartner of a muon, the stau, which is superpartner of a tau lepton and the sneutrino, which is superpartner of a neutrino.  The 1/2-spin superpartners of the Bosons, the force particles, are for the photon the photino; for the gluon, the gluino; and for W and Z bosons, the wino and zino respectively.  The hypothetical graviton's superpartner is the gravitino, and for the Higgs boson, it is the higgsino.
But; is there any evidence for supersymmetry?  Some issues in the Standard Model are resolved if supersymmetry is included.  The standard model requires input from experimental data that is "fine-tuned" to incredible precision in order to cancel "quantum jitters".  By having bosonic and fermionic particle pairs, the cancellation occurs naturally, avoiding the need for this fine tuning.  Supersymmetry also comes to the rescue of Grand Unification.  We have seen how electromagnetism and the weak force become the electroweak force at temperatures around 1015° K, and the strong force joins them (hypothetically) at about 1028° K.  However, while the strengths of the forces almost become equal, it is not perfect unless super symmetry is included.  Finally, a widely accepted theory postulates that it is the Higgs boson, permeating all of space, that gives particles their mass.  The mass of the Higgs, however, tends to become huge due to Quantum jitters, almost the Planck mass, unless supersymmetry is considered when its mass becomes about what the Standard Model requires.  All this is a strong evidence in favor of supersymmetry.
Finally, on a cosmological level, some cosmologists have proposed that the lighter supersymmetric partners are the best candidates for the dark Matter that permeates the Universe.  We shall see.  Here is another link to a discussion of supersymmetry