"Core Theory", which is to say the Standard Model of Particle Physics and General Relativity (with a cosmological constant), have only a few issues where there are serious tensions with the data or other fundamental problems.
1. GR and the SM are theoretically inconsistent.
First, the quantum mechanical, probabilistic Standard Model of Particle Physics has theoretical inconsistencies with the classical, deterministic theory that is General Relativity with a cosmological constant.
Most likely, we need some sort of quantum gravity theory to solve this problem.
2. Dark matter phenomena don't have a widely accepted specific explanation that works in all circumstances.
Second, neither General Relativity as conventionally operationalized, nor any of the fundamental particles of the Standard Model, nor composite particles made of them, can explain the phenomena attributed to dark matter.
All specific explanations of dark matter phenomena that have received intense examination from multiple independent investigators have serious flaws, and many potentially plausible explanations have not yet been seriously vetted.
(Note that "dark energy" in contrast, is satisfactorily explained by the cosmological constant of General Relativity which is part of exiting "Core Theory", although generalizing the cosmological constant to a quantum gravity theory is particularly challenging and there are challenges to this explanation.)
There are proposals (most notably this one) that potentially provide a road map to resolving these first two problems.
But no proposal that could solve all of the dark sector phenomena issues have wide acceptance. In particular, the LambdaCDM cosmology, which is the so called "Standard Model of Cosmology" has many well know flaws that make it impossible to reconcile with all of the observational evidence.
Fortunately, physicists are devoting a great deal of effort on myriad experimental and theoretical fronts from particle accelerator experiments, to cosmic ray analysis, to gravitational wave detectors, to photon astronomy at every possible wavelength, to neutrino astronomy, to analytical studies, to many body simulations, to consider every possible explanation for dark matter and dark energy phenomena.
Overwhelmingly, these experiments and analyses are ruling out possible explanations, and narrowing the parameter space of possible solutions within various leading paradigms.
It isn't a very efficient process. Every week, many proposals that are already overwhelmingly disfavored by existing observational data and analysis are introduced unaware of what came before it in slightly different subfields of the collective scientific effort to understand dark matter and dark energy phenomena. But I am nonetheless encouraged that the firehose of new data will help to bring us closer to the truth eventually.
Many proposals (although none that I find very plausible) would require significant beyond the Standard Model physics with at least a new fundamental fermion or boson that constitutes dark matter and a new fundamental force mediated a massive carrier boson that governs the self-interactions (and possibly also ordinary matter interactions) of dark matter.
3. Are lepton universality violations real, and why or why not is this the case?
Third, lepton universality violations observed in a couple of kinds of B meson decays, in which different flavors of charged leptons (electrons, muons and taus) behave differently in ways not obviously due to their rest masses alone, seem to be inconsistent with the Standard Model.
This is really the only truly significant anomaly in high energy physics that could plausible give rise to new physics that is known today.
The fact that heavier charged leptons are more scarce than lighter ones is itself notable, and is suggestive of potential Standard Model solutions. So is the fact that experimental tests in a great many other contexts do no show lepton universality violations. In other words, I think that there is most probably something subtle wrong with how the Standard Model prediction is modeled that doesn't reflect something prosaic that the experimental measurements are seeing.
There are many very exotic explanations of this phenomena that have been proposed. But, I still expect that there are much better than even odds that a Standard Model explanation to it can be found.
Also, even if new physics are required to explain it, these new physics, much like the new physics involved in inserting a CP violating phase into the CKM and PMNS matrixes when CP violation was seen experimentally in those processes, will probably be quite prosaic and won't introduce new kinds of exotic particles, like leptoquarks, nor will it fundamentally reshape the basic structure of the Standard Model.
In the Standard Model, the interactions in which lepton universality violations arise are W boson mediated interactions that would probably be most parsimoniously explained by adding to the experimentally described properties of the W boson that already defies other rules, like CP conservation and conservation of quark flavor and lepton flavor, which are observed in all other processes.
Pretty much all other anomalies in modern physics (from the muon g-2 anomaly which is probably due to an incorrect calculation of the theoretical value by some groups with a correct calculation proposed in 2020, to the hypothetical X17 boson, to the 750 GeV anomaly, to sterile neutrinos in connection with the reactor antineutrino anomaly, to anomalies in very high energy cosmic ray decays, to the muonic proton radius puzzle, to the superluminal neutrinos that the Opera experiment thought that it saw, to the anomalous neutrinoless double beta decay observations of the Moscow experiments, to the XENON 1T anomalies) have good explanations due to experimental errors or flawed theoretical predictions that don't involve new physics. Some of the anomalous experimental results have been convincingly ruled out or discredited by multiple new experiments.
The lepton universality violations and the tensions that have been resolved aren't the only experimental tensions in the Standard Model of Particle Physics.
But others tensions that remain (like a lack of perfect unitarity in the CKM matrix, or the discrepancy in the lifetime of the neutron when measured in two different ways) are widely believed to be cases of experimental uncertainty and systemic error in experiments, rather than the result of "new physics."
Real Unsolved Questions
This doesn't mean that every single unsolved question in fundamental physics has been answered.
There are several properties of neutrinos that haven't been measured very accurately and we are still unclear if the rest mass of neutrinos arises in the same way that it does for other Standard Model fundamental particles. I very much doubt that see-saw models of neutrino mass that garner so many publications are right, but I don't have an explanation of my own that is a clearly superior alternative.
We are still quite fuzzy about what triggers the "collapse of the wave function" in quantum mechanics.
We still aren't clear regarding what axioms about the nature of physics are invalidated by the quantum phenomena known as "entanglement." We instead have a trio of assumptions (commonly called locality, causality, and reality), one of which must be false.
Similarly, the fact that the correct evaluation the path integral of the propagator of quantum electrodynamics requires the consideration of hypothetical photon paths that involve photons traveling at slightly more or less than the speed of light, contrary to special and general relativity, is a mystery that could be a clue to some deep insight into the nature of the universe. This is arguably the best hint that space-time itself may not be smooth and continuous in the true quantum gravity theory. The shut up and calculate school of physics doesn't really care. But I find it one of the most intriguing aspects of the established laws of physics.
We still don't know why the experimentally measured constants of the Standard Model take the values that they do, even though it seems to be due to some kind of deeper theory, rather than being random. I have ideas culled from the literature, and there are theoretical proposals that attempt to explain them. But honestly, this line of theoretical research in fundamental physics that is probably receiving too little attention.
Lepton universality violation and all of the real unsolved questions above show strong signs of being answerable more or less entirely within the electroweak sector of the Standard Model, which is better understood and is possible to calculate more precisely with, and this is encouraging.
We have also probably devoted too little attention to the question of how plausible, minimalist quantum gravity theories impact the renormalization and beta functions of Standard Model experimentally measured constants at different momentum transfer scales. Any extrapolation of physics at the energies we've measured to the extreme high energies of the GUT scale or thereabouts that fails to take this into account must surely be incorrect, and quite modest tweaks to the Standard Model beta functions could significantly impact gauge force unification expectations, for example. In theory, this is a deterministic enterprise once the quantum gravity theory is well specified, so there is hope that is question could be solved in a single paper or two from a committed researcher.
We are still unclear about just what happened in the first moments after the Big Bang to give our modern universe the overall structure and global properties that is displays like the aggregate baryon number of the universe, the aggregate lepton number of the universe, the baryonic matter-antimatter asymmetry of the universe, and other large scale structure properties of the universe that we observe (e.g. properties often attributed to cosmological inflation).
We also haven't characterized sphaleron processes in particle physics (the only Standard Model process that doesn't conserve lepton number and baryon number) very well. The answer is almost surely not, as almost every major paper on the subject suggests, either of the two CP violating processes in the Standard Model (W boson mediated changes in quark flavor and neutrino oscillation).
I am highly skeptical the either cosmological inflation or beyond the Standard Model means of violating lepton number or baryon number are real. But I'm open to considering evidence to the contrary.
Finally, there are lots of complex processes (e.g. hadron structure, jet decays, and patron distribution functions), most of which involve quantum chromodynamics (QCD) strong force calculations that are hard to even approximate answer too accurately. These should, in theory, have an exact explanation in the Standard Model, and almost all physicists are confident that they do, but these questions are too difficult for us to currently calculate answers to with anything approaching the precision with which we can measure these complex processes experimentally. These questions will take centuries to fully master, if that ever happens.
But none of these other true unsolved problems in physics necessarily requires "new physics" beyond the Standard Model and general relativity to explain. It isn't impossible that they do, but there is no positive observational evidence, let alone unequivocal evidence, that this is the case.
Fake Unsolved Questions In Physics
Other so called "unsolved problems" in physics, like "the hierarchy problem", "naturalness", the "strong CP problem", baryogenesis, and leptogenesis, don't really deserve the name and are simply observationally unsupported presumptions about the laws of Nature that are false.
The Junk Heap Of Failed Theories
Despite the fact that we have very little apparent need for them, a huge amount of theoretical and experimental effort in fundamental physics is devoted to exploring one or more of a great many beyond the Standard Model theories.
The resolution of various particle physics anomalies is increasingly ruling out all or most of the parameter space of these theories, leaving them with little or no positive observational motivation.
These include supersymmetry, multiple Higgs doublet theories, myriad grand unified theories and theories of everything, technicolor, extra dimensions of space-time, leptoquarks, sterile neutrinos, string theory, and more.
A resolution of dark matter phenomena without new particles and of lepton universality violations without new physics would wipe the field of almost all of the rest. I think that this could happen within my lifetime (at 50 years old) or at least, within my children's lifetime. We may not be in the promised land yet, but we can glimpse it in the distance from the mountain tops.
As I explained, above, this wouldn't be a true "end of science" as there would still be questions to be asked and complex phenomena to be understood. But we are very close, I think, to having a complete and accurate set of laws of nature and could get there soon.
When we get there, I suspect that the laws of physics will again look far simpler and more elegant, and will be easier to study as well, without having to master a host of hypothetical conjectures with a sophisticated and hard to master body of research associated with them, that do not pan out.