In 1983, Milgrom came up with a simple empirical formula (MOND for Modified Newtonian Dyanamics) that reasonably successfully fit essentially all galactic rotation curves, both of all types already observed and of all future types that would later be discovered, with a single empirically parameter, a0 with a value of approximately 1.21 * 10-10 m*s-2, having the dimension of acceleration, approximately representing the gravitational field strength at which gravity appears to display an ordinary 1/r2 relationship to the ordinary luminous matter in the galaxy, and the point at which it appears to display a modified 1/r relationship to that matter.
This constant has subsequently been observed to be on the same order of magnitude as the speed of light times the Hubble constant, and as the speed of light times the square root of a quantity equal to the cosmological constant divided by three.
Subsequent research have shown that while the relationship accurately predicts the behavior of galactic scale systems quite well over a wide range of scales, that the simple MOND relationship underestimates by a factor of several times the MOND predicted amount in galactic cluster systems. Milgrom's colleague Bekenstein has successfully generalized the core observation of MOND (which merely generalized Newtonian gravity) in a modification of general relativity called TeVeS (for tensor-vector-scalar gravity). The formula for interpolating between the Newtonian and MOND regime has not definitively established empirically, and there is significant uncertainty in the measured value of the MOND constant a0 that approaches almost 50% in measurements derived from the most extremely dark matter dominated HI type dwarf galaxies.
Alternately, some of the apparent uncertainties in the fitting of the MOND constant to these extreme rotation curves could derive from the use of an insufficiently sophisticated interpolation function between the two gravitational force law regimes in a rare context that is actually sensitive to the details of this part of the naive MOND theory which has never claimed to have the interpolation function right, because the transition take place more slowly in these systems than in most other galaxies.
MOND and variants on this theme are an alternative to the dominant paradigm that the effects it observes are the product of "dark matter" which is made of particles with properties different from matter made of ordinary atoms. In the dark matter paradigm, these particles (or several types of particles) interact via gravity, but lack electric charge, do not interact via the strong force, and interact via the weak force no more strongly than neutrinos or not at all, although most dark matter particle theories assume that there is some force other than gravity that couples to dark matter. As a result, dark matter is assumed to be nearly collisionless. The standard six parameter lamda CDM cosmology model assumes that dark matter is a thermal relic comprising approximately 27% of the mass-energy of the universe and that it must be made of particles significantly more massive than neutrinos, while just 5% is attributable to ordinary matter (the balance is "dark energy", i.e. due to the cosmological constant). Newly conducted precision observations of a relatively nearby galactic cluster observationally confirms the widely adopted hypothesis that dark matter phenomena are consistent with a pressureless fluid from a general relativity perspective.
Early dark matter models did a dreadfully poor job of accurately predicting the amount of inferred dark matter that would be present in newly measured types of galaxies, and involved many adjustable parameters to describe observed dark matter halo distributions, in addition to requiring the invention of a new kind of beyond the Standard Model dark matter particle of a type never observed (even indirectly, for example, in the form of missing mass-energy) in high energy physics experiments. Of course, without a theory to explain why dark matter was distributed as it was, any dark matter theory is incomplete.
Simple N-body simulations derived from first principles from cold dark matter models such as the Navarro-Frenk-White (NFW) dark matter halo model conceived in 1997 produce "cuspy" dark matter halos that contradict the observational evidence regarding the inferred shape of dark matter distributions. Generically, cold dark matter models produce cuspy halos, and an excessive number of dwarf galaxies, both contrary to the empirical evidence, in computer simulations applying them. Warm dark matter models, with dark matter particles that have masses on the order of 2 keV rather than 10 GeV or more, appear to be more successful in these respects. But, recent N-body simulations of simple warm dark matter models still display cuspy halos inconsistent with observational data and similar to less exaggerated NFW halos. Simple warm dark matter models do not solve this problem and viable models may require some form of feedback from ordinary baryonic matter in the galaxy or self-interactions within the dark sector, for example, to replicate observed results.
Since Milgrom published his MOND theory, dark matter theories have come a long way, as exemplified by a January 22, 2014 pre-print by South African physicists Toky Randriamampandry and Claude Carignan, entitled "Galaxy Mass Models: MOND versus Dark Matter Halos." Their paper compares the fit of a pseudo-isothermal dark matter halo density distribution models with two free parameters to galactic rotation curves to that of similar calculations using one parameter MOND models for "15 nearby dwarf and spiral galaxies" that can be measured with unprecedented precision.
This study, and a similar previous one with an overlapping data set, finds that MOND produces good fits (chi squared less than two) for only 60% to 75% of the sample (differences that are probably not statistically significant given the small sample sizes and slightly different methodologies used). It also finds that the two parameter dark matter model, using fixed mass to light ratios established by stellar population models, produces slightly better fits for this data set biased towards the extreme of the range in which MOND fits are possible if the MOND constant can be individually adjusted for each galaxy and/or best fits within margins of error for other relevant measurements of the galaxies to be fit such as distance from Earth are permitted. And, this dark matter model produces significantly better fits than those made simply using the most widely accepted estimate of the MOND constant. Prior studies have produced mixed results on the question of whether "galaxies with higher central surface brightnesses tend to favor higher values of the constant a0," a relationship that the current paper's sample tends to weakly confirm.
Equally interesting, this paper confirms that the two free parameters of a pseudo-isothermal dark matter halo model, core radius and central halo density, are not independent of each other and that the surface density of the halo which is calculated as the product of these two parameters is approximately constant with a value of about 120 Mp/c2 (two previous observational estimated the value of this constant at 100 and 150, respectively). Thus, three decades later, dark matter theorists have finally produced an empirical single free parameter dark matter halo model, with a constants whose value is known to a similar precision to that of MOND's single parameter, and which produces comparable or even moderately better fits to galactic rotation curves. Of course, like MOND, the pseudo-isothermal dark matter halo model is still not derived from first principles based upon a particle with particular properties, its constant is not very precisely determined empirically, and still requires one element of new physics beyond the Standard Model and general relativity.
While these researchers, in my opinion, fail to show as convincingly as they claim, that there are now dark matter models that are clearly superior to MOND in the arena where it has historically been strongest empirically, they do demonstrate that there are finally, three decades later, dark matter models comparable to MOND in parsimony and predictive accuracy.
Despite these strong results, however, there is no consensus regarding the right way to model dark matter halos and there is considerable uncertainty regarding how to fit the observational data to any particular model.
On the other hand, some of this lack of consensus has more to do with the participants than the state of the empirical evidence. For example, an atrocious Snowmass preprint on Planning the Future of U.S. Particle Physics in Chapter 4 (The Cosmic Frontier), treats all manner of dark matter candidates which are overwhelmingly disfavored by existing combined observational evidence from astronomy, direct dark matter detection experiments (such as the Xenon experiment) and HEP experiments, such as black hole, WIMPzilla, Q-ball, WIMP Cold Dark Matter, and Stangelet Cold Dark Matter as credible dark matter candidates, while glossing over more viable possibilities like sterile neutrino warm dark matter. They also exaggerate the necessity of hypothetical axions, proposed by Pecci and Quinn in 1977 to solve the strong CP problem (which is ultimately a "why" problem within the Standard Model, and not a problem of the Standard Model failling to accurately model reality), while failing to even acknowledge decades of fruitless axions searches that should have revealed axions whose properties fit more naive expectations regarding this hypothetical particle. Much of this appears to flow from an ideological commitment to supersymmetry theories that do not provide any realistic dark matter candidates which are consistent with observational evidence since HEP experiments already largely rule out any SUSY dark matter candidates that are light enough to be consistent with the astronomy data.
The reality that Snowmass participants are loathe to admit is that many costly existing or imminent dark matter experiments designed to measure dark matter phenomena have exceedingly weak prospects for success because data obtained by other means already strongly disfavor the existence of a dark matter candidate of the type that the experiment in question could detect if it existed. In short, these big projects have been rendered obsolete before ever reporting results.