We know that it (graviton) should have the following properties: it should be massless particle because it mediates infinitely ranged force (gravity), thus it should move with the speed of light as any other massless particles do. It should have spin +/ 2 in order to mediate attraction only (no repulsion).
Graviton should have extremely small energy because according to GR (gen relativity)gravitational waves carry very small power. Thus to detect gravitons we have to have extremely sensitive devices (which we do not have) or extremely violent source of gravwaves nearby (merging black holes). So far we are not lucky with such event either.
Indirect evidence of gravitational waves (thus, gravitons) comes from all corners of GR. According to GR gravitational radiation propagates with speed c. Observation of spiraling neutron binaries (heavy compact objects for which gravitational radiation is not negligible) shows that they indeed lose energy in the amount predicted by GR: P=(32G/5c^5)(m1m2/m1+m2)^2R^4w^6 and because there is a speed of gravwaves c in this equation is in 5th power, then this radiation is very sensitive to this speed. Measured power of gravitational radiation by spiraling binaries correspond to speed of gravwaves to be equal to the speed of light within 1% experimental error.
All effects and predictions of GR depend on this speed, and if it is NOT equal to the speed of light, then all GR predictions/calculations are incorrecrt: GPS system can not accurately calculate positions in the gravitational field of Earth, gamma rays falling on Earth do not gain gravitational blue shift (as was experimentally observed back in 1961 using Moessbauer effect), planetary orbits do not precess, atomic clocks in Denver do not run faster (due to less gravity) than clocks in Washington DC, and many other confirmed facts.
Exchange by virtual gravitons (they call them scalar gravitons in theoretical physics) in 3D space mathematically results in what we call Newton's universal law of gravity (inverse square law).
So, basicly question is not if gravitons exist but how to detect their extremely weak energy flow (gravitational radiation) with our very rough devices.
By the way, in string theory graviton emerges naturally as a quantum fluctuation of any energy. And it has spin 2.
