The most challenging physical phenomena What are examples of endeavors, in the history of mankind, to understand physical phenomena with models which were proved to be incorrect  later, reformed significantly,  or are still under development?
 A: There were many models overturned throughout history, I will list some of the most salient ones. I will ignore the ones that predate modern science, the most prominent one being the geocentric model of the solar system, and I will confine myself to wrong ideas that were scientifically accepted as probably true at some point in history.

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*Phlogiston: This is the fluid that carries heat. It was imagined to be a quantity, like electric charge, that is conserved. Experiments with cannon-boring machines showed that the amount of heat that you can make by doing mechanical work is only limited by your patience and force, so that the phlogiston fluid was discredited. This work is associated with the name of Joule. The replacement for Phlogiston is the modern notion of energy, which includes both mechanical and heat energies as one, and heat is no longer a quantity conserved separately from mechanical energy.

*Lumineferous Ether: This is the idea that light propagates in a medium, which fills all of space, and picks out a preferred frame, in which the speed of light is equal in all directions. Relativity showed that such a frame does not exist--- that light moves in the same speed in all directions regardless of the motion of the source or the observer. This new symmetry removed the need for a non-relativistic light ether. Modern etherlike ideas are relativistically invariant, and include the QCD condensates and the Higgs mechanism.

*The Drude model: When electrons were known, but quantum mechanics did not exist, Drude proposed that a current is a drift of electrons. The idea was that electrons make a gas inside a metal, and for some reason they drift along as if they were in free space. The result predicted a very slow drift velocity of electrons, the Drude velocity. Drude's model was made quantum mechanical, and this required that the electrons make a quantum Fermi gas. The Fermi gas is very cold at room temperature, since the typical temperature at which it becomes classical is of the order of the melting temperature of the metal. The behavior of a degenerate Fermi gas explained the specific heat of metals, and gave a correct velocity for the current carrying electrons.

*The ether-knot theory: This idea was due to Kelvin, that atoms are vortices in the ether, different atoms are different kinds of knots, and molecules are links. It was very popular at the turn of the century, because it could explain why atoms were discrete, but was shown to be wrong when Bohr's atom qualitatively and quantitatively explained the periodic table and the spectrum of Hydrogen.

*The Plum-pudding model: This imagined that electrons were embedded in a big diffuse positive charge, which was the atom. This theory predicted why atoms can have special resonance frequencies where they scatter light strongly. These frequencies were the resonant motions of the electrons inside the plum-pudding. The theory is incorrect, and the correct laws of resonant scattering are only provided by quantum mechanics.

*The Bohr-Sommerfeld quantum theory: This is the old quantum mechanics, where the action was given integer values. It was an important intuition for developing modern quantum mechanics, but it is now recognized as just an approximation to the actual quantum mechanical laws. The modern quantum mechanical laws were intuited from the Bohr-Sommerfeld laws by Heisenberg in 1925, and the same laws were derived by a different path, using wave-particle duality by DeBroglie, Einstein and Schrodinger. The Bohr-Sommerfeld theory is still a useful approximation, but it is not considered fundamental anymore.

*Bohr-Kramers-Slater theory: This idea was that electron orbits are quantized, but the electromagnetic field is not. The theory predicted that energy is not conserved. The reason is that electromagnetic waves which don't come in photons can excite electrons in quantized orbits on many atoms at once, even if there are too few photons in the wave to do that. The modern quantum theory, plus the observation of Compton scattering, demonstrated that photons are real, and killed the theory.

*Nuclear electrons: In the 1920s and 1930s, before the neutron was discovered, the mass of nuclei were known to be approximately integer multiples of the mass of the proton. Since it was considered unlikely that there would be a neutral particle with the same mass as the proton, people assumed that there were electrons tightly bound in the nucleus. This idea was inconsistent with quantum mechanics, which predicted that an electron confined to a nucleus would be about as massive as a proton, and when the neutron was discovered, the idea was jettisoned. You see the theory of nuclear electrons in old papers.

*Integer charged Sakata quarks: In 1957, Japanese physicist Sakata explained the structure of the known strongly interacting particles by assuming they were all made of the proton, neutron, and lambda baryon. This idea is roughly successful, but only because the proton, neutron and lambda are surrogates in this theory for the up, down, and strange quark. The idea was fixed up to the quark model by Gell-Mann and Zweig, but Sakata is strangely neglected in this story, perhaps because of his strong Marxist political leanings.

*Landau mean-field exponents: In the field of critical phenomena, second order phase transitions are those which have power-law divergences in the correlation length and the averaged field fluctuations at the transition. The power laws for the divergences were predicted by Landau using general principles of analyticity, later made more precise by Thom in catastrophe theory. These predictions fail, and Landau recognized that this was a sign of an important new discovery to be made. The discovery was modern re-normalization, due to Widom, Kadanoff, Wilson, Fisher and developed by many others.

*Kolmogorov Turbulence Theory: Kolmogorov proposed an approximation to turbulence which derived the energy in each mode from an argument which does not require much more sophistication than dimensional analysis. The same theory was reproduced by Onsager and Heisenberg, probably completely independently, during the WWII years. The K41 theory predicts the velocity-velocity correlation functions in fully developed turbulent flow. The theory was first thought to be exact, but in the 1960s, it was slowly shown to be only a rough first approximation, with new phenomena of intermittency altering the correlation function power laws.

*Steady state cosmology: This was the idea that the expansion of the universe is produced by a field with a positive cosmological constant, so that we live in a de Sitter vacuum. Further, as the universe expands, new H atoms are produced from the expansion so that the universe is in a steady state. This was killed by evidence for a hot Big Bang, the cosmic microwave background, plus the observation that old galaxies are noticeably different in their statistical properties from modern ones, they are immature and irregular, contrary to steady state predictions.

*Frozen star black-holes: Before the black hole theory was advanced, many people, including Einstein, proposed that something terrible happened at the horizon, which either led to blow ups in energy, or to freezing matter on the exterior. This is somewhat true, in that it takes forever for an object to cross the horizon from an exterior point of view, but the modern understanding requires that objects have an interior to go into.

*Black hole remnants: This was the short-lived idea the black holes leave behind a small point-like object with a huge entropy when they decay in the final stages. This was designed to fix the black-hole information paradox.

*Black holes make other universe: This idea was associated with the information loss puzzle--- how can black holes lose information? They must be making a new universe.

*Cosmological constant suppression due to diverging mode in quantum gravity Path integral: This idea is due to Coleman, and unlike many of his other brilliant contributions, it turned out to be wrong. The idea there was that the diverging scale-factor path-integral factor in the quantum gravity path integral leads to a zero cosmological constant. This theory was killed by the observation of a nonzero cosmological constant.

One could continue giving examples, but it is easier to look at the old literature and find all claims. Many of these claims are incorrect, and each one is usually an example of this sort. It is good to know the wrong turns, so as not to rediscover an old new idea.
For the most challenging physical phenomena, those which are currently the most difficult to understand, I will have to choose one of the many mysteries. I would probably say:

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*High temperature Superconductivity: What makes ceramics superconduct? It is clearly a purely electronic thing, nothing to do with phonons, but the attraction mechanism is not completely understood. This has been an active subject for 20 years, but I do not see a good answer in the literature.

*Regge theory: How do you produce a Regge trajectory from a confining field theory? This question lies at the intersection of string theory and quantum field theories, like QCD.

*How do clouds form? How do they separate charges? The phenomena in clouds is completely ill understood, and important for giving climate science more precise predictions. There are many approximate models here

Again, there are too many to list, you just explore the literature and find open questions.
