For all of those of you who haven't been paying attention to physics for the last century or so, here is a quick recap of the the current situation from the introduction to a recent pre-print on quantum gravity:
The past few hundreds of years of development have combined precision experiments with abstract theoretical reasoning, and our current understanding of the basic building blocks of nature is as follows.
The universe contains matter, which is acted upon by forces.
All observed forces are consequences of only four fundamental forces.
Three of these - electromagnetism, the weak nuclear force, and the strong nuclear force - are described by the Standard Model (SM) of particle physics, which also lists all known matter particles – quarks and leptons – that combine to make the various composite particles observed in nature.
All forces and matter types in the SM are described by fields filling all of spacetime. The equations describing these fields include the effects of quantum mechanics and relativity, so that the SM is an example of a quantum field theory (QFT).
Indeed, the particles themselves emerge as quanta of the fields, with the canonical example being the photon (a quantum of the electromagnetic field). Furthermore, the fields have certain abstract mathematical symmetries, so that this type of QFT is also called a (non)-Abelian gauge theory.
There. Got it. The Standard Model has been in place more or less since the 1970s, and general relativity is more or less unchanged from its state a hundred years ago.
Recent Standard Model Developments - The Higgs Boson and Neutrino Physics
The only really significant developments in the Standard Model since the 1970s have been the discovery of various Standard Model predicted particles (the last of which was the Higgs bosons discovered in 2012 completing the set) and some properties of neutrinos.
We've learned that neutrinos are massive and oscillate between mass eigenstates according to a three by three complex matrix with four parameters known as the PMNS matrix. We know the difference between the masses of the three mass eigenstates, but not their absolute values (although we have capped the maximum value of the sum of the three masses) or the order of the masses, and we know three of the four PMNS matrix parameters, although not as precisely as we'd like and not the fourth CP violating parameter. We also don't know which of two mechanisms generate the neutrino masses.
Dark Matter and Dark Energy
On the gravitational front, we've learned that we need a cosmological constant or dark energy or the equivalent, and that we need either dark matter or a tweak to the behavior of gravity in weak gravitational fields to explain a lot of astronomy observations. Unlike the remaining aspects of the Standard Model which require us to simply set a few parameters, the dark matter/dark energy problems are truly unsolved.
It seems like it ought to be possible to formula General Relativity as a quantum theory (it is currently a classical theory like Newtonian physics) and that quantum theory might even resolve some other loose ends. But, we haven't managed to do that yet and our two "Core Theories" are theoretically inconsistent without it. This may or may not be related to the dark matter and dark energy problems.
The First Moments After The Big Bang - An Optional Problem
There is also some lack of clarity regarding just what happened in the first few moments after the Big Bang - primarily baryogenesis, leptogenesis, cosmological inflation, and matter-antimatter asymmetry. But, since these things happened at basically unreproducible energy scales, there is really no practical or functional reason that we must know the answers to any of these questions which together conspire to create initial conditions a very short period of time after the Big Bang, after which conventional physics takes over.
Within The Standard Model - An Optional Problem
We would also prefer to condense the somewhat parameters heavy, multi-equation "Core Theory" to an even deeper "within the Standard Model" theory or even a "Theory of Everything", but ultimately that would be simply an academic exercise once finished and wouldn't have observable differences from the Standard Model and General Relativity with the parameters it predicts.
Step back one more step and there are some more practical, probably not fundamental, issues left.
We still don't really understand how scalar mesons and axial vector mesons form or take the mass spectrum that they do. We are still trying to better grasp the physics of glueballs and hadrons with more than three constituent quarks. QCD calculations are very hard to do precisely, taking lots of time, expertise and computational capacity to conduct. There are quite a few Standard Model and General Relativity constants that aren't measured terribly precisely and we could do better.
There are also a few relatively minor glitches in comparing experiment and theory. Most notably: there are some indications of a violation of charged lepton universality, a measurement called muon g-2 does not quite square with the predicted value, the charge radius of muonic hydrogen isn't quite right, and the predicted abundance of Lithium-7 in the universe isn't quite right.
Dreamers hope these are signs of new physics. Cynics think this is probably due to a combination of measurement errors and flawed calculations of the theoretically predicted values in subtle ways.
We would like to understand black holes, galaxy formation and galactic cluster formation better.
Lots of these problems will take more experiments, observations and calculations to solve, but are in principle eminently solvable. A handful of mostly optional problems may be a bit more intractable.