Galactic rotation curves, gravitational lensing, and the kinematic behavior of galactic clusters all point to the existence of substantial amounts of dark matter beyond the matter that can be inferred from the light we detect from stars and the black holes that we infer are at the centers of galaxies and certain other systems.
But, the dominant cold dark matter paradigm has pretty much been disproven by observation and experiment. Cold dark matter doesn't arrange itself the way it needs to in order to produce the observed results.
Well, dark matter is one way to reconcile what we see with the models of gravity we use to predict how these systems will behave. Basically, at the galactic scale the field strength of gravity does not fall off in a 1/r^2 manner the way that it should from the matter distributions that we can observe or infer from patterns of starlight we can see. After a critical value of gravitational field strength called a0, we observe galactical scale gravity falling off at a 1/r rate instead. We see the same thing in wide orbit binary stars.
A theory called MOND simply applied the phenomenological gravity law that this was the case and was able to predict almost all galactic scale gravity behavior, predict the amount of dark matter present in previously untested systems and its distribution, and the wide orbit binary star behavior with accuracy with this one constant and a simple modified Newtonian gravity equation.
The simple version of MOND is inconsistent with General Relativity where it matters, but it succeptible to being generalized to a formula that reduces to General Relativity. Several similar theories have also been proposed.
But, maybe General Relativity is good enough on its own. A recent discovery that the amount of ordinary matter in eliptical galaxies has greatly underestimated reduced the amount of dark matter to be explained from a roughly 3-1 dark to visible ratio to a roughly 1-1 dark to visible ratio. And, most early models at the galactic scale simple used Newtonian gravity without corrections for General Relativity.
Efforts to describe galaxies with full fledged General Relativity have revealed that much of the unexplained dark matter that seems to be there when a Newtonian approximation is used disappears when full fledged General Relativity is used. Not all of it, but a lot of it.
Also notably, the large scale structure of the universe appears to be indifferent to the proportions of dark matter and ordinary matter contained in them.
Some basic intuitions, stated in less technical form, about what is going on, follow:
1. Gravitational effects in General Relativity are a function of five vectors and a scalar that go into the stress-energy tensor. Both General Relativity and Newtonian gravity reference the scalar of rest mass. But, General Relativity adds three element vectors (corresponding to the three dimensions of space) all of which are derivatives with respect to time, for (1) linear momentum, (2) angular momentum, (3) special relativistic boost momentum, (4) pressure, and (5) energy flux.
2. In galactic cases, where MOND works, it is possible to choose a frame of reference where linear momentum is zero or nearly zero, the speeds involved make the special relativistic boosts trivial relative to other factors even in the weak field, pressure is nearly spherically symmetric, and energy flux (mostly from emitted photons from stars) is nearly spherically symmetrc. But, angular momentum is not spherically symmetric in a galaxy or wide orbit binary star and makes a non-trivial contribution relative to rest mass. Angular momentum is basically all in a two-dimensional circular direction in a galaxy or binary star system, so it make sense that unlike the spherically symmetrical and scalar parts of gravity, which fall off as 1/r^2 which corresponds to the surface of a sphere (4/3(pi)r^2) that angular momentum effects would fall of as 1/r which corresponds to the circumference of a circle (2(pi)r).
3. a0 makes sense if the structure of the systems where MOND works have a similar relationship between mass distribution with Newtonian gravity relative to angular momentum relative to the component of gravity in General Relativity due to the angular momentum component of the stres-energy tensor. Some of this flows from the formulas for angular momentum and general relativistic gravity, but some of this flows from the dynamical process by which galaxies form which shows strong evidence of structure (there are two main distinct types - spiral and elliptical, they have non-overlapping mass ranges, they have similar structures within each category, there is a maximum galaxy size).
4. A phenomenological test that would distinguish dark matter and MOND theories which are generically spherically symmetric, from full general relativity calculations (too complex for anyone to have done exactly), is that in a general relativity extension, MOND/Dark matter effects should not be visible in parts of the weak gravitational field that are at a high azimuthal angle relative to the two dimensional plane in which the galaxy rotates. Neither MOND theories nor dark matter could explain a discrepancy of this type with any confidence, while general relativity could. It ought to be not terribly difficult to make a few observations along this line.
5. The exceptions to MOND success are also suggestive. The Bullet Cluster, which is the classic counterproof of MOND-like theories, is the only really decisive example of a gravitational system where linear momentum is important in every frame of reference relative to rest mass and angular momentum, complicating the analysis in full general relativity greatly. Galactic clusters have two layers of angular momentum at work - intragalactic and the angular momentum involved in the relative motion of the different galaxies in the cluster, and may have unusually matter density profiles that give rise to asymmetric pressure profiles. There is no nice neat pattern for galactic clusters as there is for galaxies themselves so the analysis is more involved.
Of course, it would also be quite impressive if General Relativity, almost a century after it was formulated in 1916, was discovered through better calculations of its effects in more complex systems to account for both dark matter in the absence of quantum gravity corrections, as well as accounting for dark energy as it already does (via the cosmological constant) with the dark matter proving to be simply a huge theoretical error and an ordinary matter counting shortfall. And, this would eliminate the need for particle physics to discover some dark matter candidate that could fit the bill, which would be nice, because particle physics is running out of options that fit the experimental data that put a number of constraints of dark matter's behavior. This would make the agenda for quantum gravity a quite modest one; we would only need it where quantum mechanics and general relativity are both relevant.
Even if unaccounted for ordinary matter and the non-Newtonian effects of General Relativity don't eliminate the need for all dark matter to explain what is observed, a reduced quantity of dark matter to account for could ease strains on particle physicists to find a plausible dark matter candidate and strains on cosmologists to figure out where the dark matter could have come from after the Big Bang.