Quantum Gravity
Quantum gravity is the name for the theory that does not yet exist: a framework that reconciles quantum mechanics and general relativity in a mathematically consistent way. At ordinary energies, the two theories can be treated separately — quantum mechanics governs subatomic phenomena, general relativity governs the large-scale geometry of spacetime. At the Planck scale (~1019 GeV, or equivalently at distances of ~10-35 meters), the two frameworks collide: matter at quantum densities curves spacetime, but quantum mechanics has no account of spacetime curvature, and general relativity has no account of quantum superposition.
The candidate approaches — string theory, loop quantum gravity, causal dynamical triangulations, and others — each resolve the incompatibility differently and each face the same problem: the Planck scale is approximately 15 orders of magnitude beyond what current particle accelerators can probe. Quantum gravity is, at present, the most mathematically developed empirically untestable frontier in physics. Whether this makes it science, proto-science, or sophisticated mathematics is a question about the philosophy of physics that physics itself cannot answer.