Here is a selection of problems that are currently of interest to workers in the field of black hole accretion. These are problems that particularly interest me, and I make no claim that they are widely agreed upon as among the most pressing issues.
While there is fair agreement about the nature of each XRB state, the factors that cause transitions between states are much less clear. In particular, the transition from the thin disk state to the hot, radiatively inefficient (and presumably two-temperature) state, is poorly understood. Both states are possible for a range of accretion rates spanning many decades, and there is a large "potential barrier" against converting a thin disk to a hot torus, because the high density of the thin disk keeps electrons and ions extremely tightly coupled. Some kind of bootstrap evaporation process may be necessary, as has been proposed for cataclysmic variables.  Formation of coronae above thin disks has been observed in magnetohydrodynamic simulations with radiative cooling, but the effects of microphysical plasma processes (such as viscosity and electrical resistivity ), which may regulate the level of magnetic activity, have yet to be assessed.
The question of what triggers state transitions is tied up with the question of what causes the hysteresis (Fig. 2), in which the transition from a hard (radiatively inefficient) to a soft (radiatively efficient) state occurs at a higher luminosity than the reverse transition. Apparently there is a second parameter (in addition to / E) affecting the transition, and this parameter is correlated with location in the cycle. The magnetic flux threading the disk, which can depend on accretion history, is an attractive candidate, [66, 12] as is the magnetic Prandtl number, the ratio of viscosity to resistivity.  Other, less orthodox mechanisms, such as the history of disk warping, may also play a role. 
The simple α-parameterization of angular momentum transport in thin accretion disks  predicts that disks dominated by radiation pressure should be thermally and viscously unstable,  i.e., they should heat up and thicken while clumping into rings. Yet the luminous, thin-disk states of XRBs show surprisingly little variability,  suggesting that this prediction is not borne out. This would not be surprising, given the oversimplification inherent in the α-model, were it not the case that simulations appear to show at least the kind of thermal instability predicted  (the models have not been run long enough to check viscous instability). Models to address this problem rely on providing an extra channel for release of energy, e.g., through disk turbulence  or winds , and/or diluting radiative pressure support for the disk, e.g., through magnetic fields.  Until we understand the stability properties of luminous disks, it will be hard to develop a compelling explanation for state transitions.
The Eddington limit is often regarded as an upper limit to the luminosity of an accreting black hole, and therefore as setting an upper limit to the mass accretion rate. In disklike flows with a super-Eddington mass supply, such as the XRB SS 433, there are indeed reasons to believe that most of the mass flux in excess of E is reversed and flung away before getting anywhere near the black hole. The most luminous quasars also seem to respect the Eddington limit, although it is possible that this could result from a selection effect (super-Eddington quasars might be sufficiently obscured to have escaped identification) or a coincidence due to a distribution in mass supply rates that decreases rapidly with . But we have seen that in starlike flows, such as the super-Eddington phases of TDEs, such self-regulation may be impossible. And GRBs clearly violate the Eddington limit by many orders of magnitude.
Exceeding the Eddington accretion rate should be distinguished from violating Eddington's limiting luminosity. Black holes can grow at a rate that exceeds LE / c2 for a number of reasons. For one, radiation produced in an accretion flow need not escape; it can be trapped and swept through the horizon.  This is one aspect of accretion that truly distinguishes black holes from other compact objects such as neutron stars. For another, the matter responsible for liberating most of the accretion energy is not necessarily the same matter that radiates it away. In a flow with large density contrasts on small scales, radiation could "go around" opaque clumps, escaping mainly through the low-density interstices. But if the low- and high-density regions are tied together by magnetic tension, a super-Eddington escaping flux will not exert enough force to stop accretion. [74, 75] For a third, some gas in a radiatively inefficient accretion flow could be swallowed with low binding energy, allowing a large amount of accreted matter to release relatively little radiation.
From the point of view of escaping luminosity, the Eddington limit is a spherical idealization, sensitive to geometric modifications. To give a simple example, a geometrically thickened disk, its surface a cone about the rotation axis, can exceed the Eddington luminosity by a factor proportional to the logarithm of the ratio between the outer and inner radii.  The apparent luminosity is enhanced by an additional factor because this luminosity is focused into the solid angle subtended by the disk surface. Inhomogeneities in the disk density, such as those caused by "photon bubbles," can also create intrinsically super-Eddington luminosities. 
More subtle are global effects that make accretion flows unable to regulate their power outputs, such as the steep density profiles expected to develop in starlike accretion flows. Here we have no obvious theoretical reason to demand that the flow rid itself of the excess energy in an orderly way; it would be quite plausible if the flow blew itself apart. But the evidence from super-Eddington TDEs, and possibly from GRBs, indicates that somehow a pair of jets is able to carry off the excess energy. How efficiently this kinetic energy is converted into radiation is another challenging unsolved problem.
Power density spectra of XRB variability display an array of rather narrow peaks, called quasi-periodic oscillations (QPOs), which often contain a significant amount of power (a percent or more) in all but the thermal, thin disk states. High-frequency QPOs typically have frequencies of hundreds of Hz, and are presumably associated with dynamical processes (orbital motions, p-mode oscillations, etc.) in the inner portions of the accretion flow. A 3:2 resonance which is often seen can be interpreted in terms of epicyclic motions in a relativistic gravitational field. 
Low-frequency QPOs, with frequencies of less than 0.1 to a few Hz, are more mysterious because they are unlikely to be associated with disk dynamics very close to the black hole, where most of the energy is liberated, yet they can carry a significant fraction (up to a few percent) of the total accretion luminosity. Both kinds of QPOs are primarily a feature of the hard X-ray spectral bands, which is believed to be produced by a hot, radiatively inefficient (two-temperature) accretion flow in low luminosity states, and a corona surmounting a thin, magnetically active disk in high-luminosity states. For the low-frequency QPOs, this suggests that energy is being spread through the inner hot region by nonradiative processes, to be modulated by some kind of oscillation or rotational motion at a rather sharp outer boundary. Candidates for the modulation mechanism include coherent Lense-Thirring precession of a hot accretion flow with an outer radius of several tens of gravitational radii  and rigid rotation of an extended (∼ 103 Rg) magnetosphere,  which requires an enormous amount of trapped magnetic flux. Other suggestions appeal to thermal or viscous timescales, but no model has been particularly satisfactory to date.
According to standard theories for jet formation, the power of a jet depends on the total magnetic flux threading the engine, LJ ∼ Φ2 Ω2 / c, where Φ is the net magnetic flux and Ω is the angular velocity of the crank. While the rms magnetic field strength in the inner region of a TDE accretion flow can be substantial, most of this is associated with turbulent field resulting from the MRI. The net poloidal flux is limited to the magnetic flux contained in the disrupted star, which is too small to power the observed jets by about five orders of magnitude.  Whether a starlike, super-Eddington accretion flow can generate large-scale fluctuating fields with enough coherence to power the observed jets is an open question. GRB jets face a similar shortfall, although the likely discrepancy is only one or two orders of magnitude.
This suggests that jets in TDEs and perhaps GRBs are propelled by the energy in chaotic magnetic fields,  which would quickly decay by turbulent reconnection into radiation pressure. Radiative acceleration of optically thin gas to relativistic velocities is severely limited by radiation drag effects, which are made worse by relativistic aberration,  but these effects can easily be ameliorated if the flow entrains a large enough optical depth in ambient matter to shield itself. A marginally self-shielded jet could be accelerated to a Lorentz factor that is some fractional power (∼ 1/4) of the Eddington ratio LJ / LE, which could explain why GRB jets are so fast.
It is curious that the most highly relativistic jets known — those associated with GRBs — should emerge from the most optically thick, radiation-dominated regions. Perhaps this reflects the difficulty that large-scale coherent magnetic fields face in converting most of their energy into motion. Once they reach moderately relativistic speeds, corresponding to rough equipartition between kinetic energy and Poynting flux, the electric field cancels out nearly all of the accelerating force and the energy conversion process stalls. Some form of magnetic dissipation may be necessary to catalyze further conversion of Poynting flux into kinetic energy. [83, 84]
These arguments also apply to the jets which are more likely to be propelled by coherent magnetic flux, i.e., those produced in AGN and XRBs. It is commonly believed that jets cannot be propelled to extremely relativistic speeds by thermal pressure, but this is true only if most of the energy is quickly transferred to electrons, which cool rapidly. At the low densities and optical depths present in AGN jets, there is no reason to exclude a sizable contribution from relativistic ion thermal pressure. On the other hand, the fact that these jets seem to be limited to fairly low Lorentz factors, in the range of a few to a few tens, might indicate that such dissipative processes are not very effective.