There are several competing approaches to unifying general relativity and quantum mechanics--string theory being far and away the most popular. One of the implications of string theory as currently envisoned is something called Supersymmetry. (I've joked that it has been a super-cemetery of careers.)
It needs a little explanation.
We have a small suite of fundamental particles, which for these purposes we divide into two types according to spin (an angular momentum that is intrinsic to the particle--you can't get rid of it, just change its direction). It turns out that the smallest unit of angular momemtum you can change is ħ=h/2Pi, where h is Planck's constant. That means that fundamental particles will have intrinsic spin in some units of ħ=h/2Pi. 0, ±ħ, ±2ħ are obvious values, but a little thought will persuade you that another possible set of values is ±ħ/2, ±3ħ/2, etc. (ħ/2 - ħ = -ħ/2).
As an example of the latter, a photon has spin ħ, and an electron has spin ħ/2. If the electron emits a photon, the size of its spin will stay the same, but the direction will be negative to what it was before. You get the same spacing between rungs of the ladder, but the origin differs.
That difference between integer and half-integer turns out to be significant--the two types of particles have a different symmetry. The most famous difference is that there can't be two half-integer (fermion) particles in exactly the same state, while there can be an infinite number of integer (boson) particles in a single state.
In quantum mechanics you must take into account ephemeral particles--a photon flying blythely along may temporarily split into an electron and positron, which recombine to make a photon again. There are differences between the contributions of fermions and of bosons, and some calculations produce infinities. I take that to mean that there's something wrong with the way thing are calculated, but I don't know what.
Anyhow, one framework that lets you do the calculations nicely, and comes up with good answers, requires that every fermion have a partner boson of otherwise similar type, and every boson a partner fermion. That's supersymmetry. I'm told it's beautiful.
If those particles exist, that's fine. Decades of looking have only found limits. One problem is that with so many new particles, you have so many new masses to tinker with as parameters of your theory. If the mass of X is bigger than that of Y you might get X decaying into Y, but if the mass hierarchy is reversed things might go the other direction.
Those masses could range from next-to-zero to almost up to the (unachievable with any foreseeable technology) Planck mass, and the theory offers no guidance.
But some SUperSYmmetry models are prettier than others, and those suggest relatively light particles--which we should have seen. But so far--crickets.
Arguments like the seesaw mechanism say that if we have neutrinos with next to zero mass and another clump of particles with mass millions of times bigger, one should have others way more massive than those in turn. (I oversimplify.) We're not likely to ever see those directly.
Arguments from beauty in the equations suggest we should have lots of those SUSY particles within -- if not easy reach, at least easy effects. Particles too heavy to create directly can still have a detectable impact on lower-energy interactions.
So far, nothing. I'm far from expert in string theory, but it seems like a very promising approach--except that it hasn't worked yet.
Everybody expected the Higgs to be there, and at roughly that mass. There were effects that suggested it was there--sort of like a mountain leaves a shadow far away from it. I was rooting for it to not be there and leave us with some new puzzles--but that's just me.
Many of us were expecting SUperSymmetric particles to appear at Fermilab, and at LEP, and at the LHC--but the limits just got tighter and tighter. There aren't any shadows of SUSY particles showing up. People have been muttering for years now.
And then there's dark matter. Maybe. If so, there are a few possible SUSY particles that might be candidates for dark matter. But we haven't seen them.
Do we keep building bigger machines to keep looking? I'm not involved in the arguments about that, but they get pretty bitter. I remember the howls when LEP was told that their hints for a Higgs weren't going to be good enough to delay the schedule to let them run an extra year. That turned out to be the right decision--the "shadow of the Higgs" they found was a statistical fluctuation and the real Higgs mass was far too large for them to have seen.
The cost of the new machines is far from trivial, and the time required to build one is a substantial fraction of a physicist's career. Nobody wants to waste money and time, and nobody wants to just barely miss the breakthrough--pick one.
UPDATE: "Limits" in this context means "We did not see any sign of particle X, and if it had a mass less than 120GeV/c^2 we're confident we should have seen it, so we conclude that if it exists, it must have a mass greater than 100GeV/c^2. Maybe we can look again with a different experiment." For me "limits" translates to "We didn't find it."