One of the earliest and initially successful alternatives called MOND by Professor Milgrom was a modified law of gravity. It had a solid predictive value at galactic scales but has crashed and burned in the galactic cluster context. The leading successor to this approach is now J.W. Moffat, of Waterloo, Ontario, whose claims that his ever name changing theory, currently known as MOG for modified gravity, overcomes problems of earlier theories with the same approach such as MOND (a good overview of both theories is found here and also at this power point presentation).
Notably, the theory identifies inertia as the cumulative gravitational effect of distant objects, an idea similar to the idea known as "Mach's principal" but with a different theoretical basis. The flavor of the theory can be seen in the early part of one of his recent papers:
The preferred model of cosmology today, the CDM model, provides an excellent fit to cosmological observations, but at a substantial cost: according to this model, about 96% of the universe is either invisible or undetectable, or possibly both. This fact provides a strong incentive to seek alternative explanations that can account for cosmological observations without resorting to dark matter or Einstein’s cosmological constant.
For gravitational theories designed to challenge the CDM model, the bar is set increasingly higher by recent discoveries. Not only do such theories have to explain successfully the velocity dispersions, rotational curves, and gravitational lensing of galaxies and galaxy clusters, the theories must also be in accord with cosmological observations, notably the acoustic power spectrum of the cosmic microwave background (CMB), the matter power spectrum of galaxies, and the recent observation of the luminosity-distance relationship of high-z supernovae, which is seen as evidence for “dark energy”.
Modified Gravity (MOG) (Moffat 2006) has been used successfully to account for galaxy cluster masses (Brownstein & Moffat 2006a), the rotation curves of galaxies (Brownstein & Moffat 2006b), velocity dispersions of satellite galaxies (Moffat & Toth 2007c), and globular clusters (Moffat & Toth 2007b). It was also used to offer an explanation for the Bullet Cluster (Brownstein & Moffat 2007) without resorting to cold dark matter.
Remarkably, MOG also meets the challenge posed by cosmological observations. In this paper, it is demonstrated that MOG produces an acoustic power spectrum, a matter power spectrum, and a luminosity-distance relationship that are in good agreement with observations, and require no dark matter nor Einstein’s cosmological
constant. . . .
2 MODIFIED GRAVITY THEORY
Modified Gravity (MOG) is a fully relativistic theory of gravitation that is derived from a relativistic action principle (Moffat 2006) involving scalar, tensor, and vector fields. . . .
2.1 Scalar-Tensor-Vector Gravity
Our modified gravity theory is based on postulating the existence of a massive vector field, μ. The choice of a massive vector field is motivated by our desire to introduce a repulsive modification of the law of gravitation at short range. The vector field is coupled universally to matter. The theory, therefore, has three constants: in addition to the gravitational constant G, we must also consider the coupling constant ! that determines the coupling strength between the μ field and matter, and a further constant μ that arises as a result of considering a vector field of non-zero mass, and controls the coupling range.
As one of his earlier papers (possibly with an earlier version of the theory, I have trouble discerning whether it is precisely the same or not) explains:
An important feature of the . . . theories is that the modified acceleration law for weak gravitational fields has a repulsive Yukawa force added to the Newtonian acceleration law. This corresponds to the exchange of a massive spin 1 boson, whose effective mass and coupling to matter can vary with distance scale. A scalar component added to the Newtonian force law would correspond to an attractive Yukawa force and the exchange of a spin 0 particle. The latter acceleration law cannot lead to a satisfactory fit to galaxy rotation curves and galaxy cluster data. . . .
A modified gravity theory based an a D = 4 pseudo-Riemannian metric, a spin 1 vector field and a corresponding second-rank skew field Bμ and dynamical scalar fields G, ω and μ, yields a static spherically symmetric gravitational field with an added Yukawa potential and with an effective coupling strength and distance range. This modified acceleration law leads to remarkably good fits to a large number of galaxies  and galaxy clusters . . . .
In contrast to standard dark matter models, we should not search for new stable particles such as weakly interacting massive particles (WIMPS) or neutralinos, because the fifth force charge . . . that is the source of the neutral vector field (skew field) is carried by the known stable baryons (and electrons and neutrinos). This new charge is the source of a fifth force skew field that modifies the gravitational field in the universe.
Translated into English, one way of articulating what this theory does is do away with dark matter, dark energy, extra dimensions, the Higgs particle or other undiscovered fundamental particles in favor of what you could either call a modification of the law of General Relativity, or a fifth force (in addition to the electromagnetic force, the strong nuclear force, the weak nuclear force and traditional gravity). His theory also identifies the Big Bang as a non-singularity from which the Second Law of Thermodynamics proceeds in opposite time directions (forward in time in ours, backward in time on the other side of the Big Bang).
Thus, overall Moffat provides a less weird explanation of the world than most other prevailing theories trying to explain current observations, while claiming to fit the data. This doesn't mean that Moffat is right, or even that I believe he is right. Minority theories in science usually fail and usually fail for good reason. But, it would certainly be comforting if he was right, and it appears that it may be feasible to figure out if he is right using means available over the next decade or two.
If the Large Hadron Collider defeats everyone's expectations and fails to detect a Higgs boson, then Moffat's scientific stock will rise immensely.
Footnote, a non-MOG paper of Moffat's about the nation that gravitons may be bound pairs to neutrinos and could explain dark energy is similarly interesting:
The graviton is pictured as a bound state of a fermion and anti-
fermion with the spacetime metric assumed to be a composite object
of spinor fields . . . . If we assume that the fermion is a light neutrino
with mass m ∼ 10−3 eV, then we obtain the effective vacuum density
¯ ∼ (10−3 eV )4, which agrees with the estimates for the cosmological
constant from WMAP and SNIa data. . . .
[W]e have predicted a vacuum density ¯ρ ∼ (10−3 eV )4 in agreement with λCDM model estimates from WMAP and SNIa data, when we identify the bound state fermion associated with the graviton condensate with a light neutrino with mass m = m ∼ 10−3 eV. By identifying ψ with a light neutrino field, we have predicted the correct magnitude of ¯ρ that fits the λCDM model interpretation of dark energy. This suggests that we describe the dark energy as graviton condensates formed from fluctuating light neutrinos. The source of dark energy would be light neutrino and anti-neutrino condensates.
The research of Jack Burns from the University of Colorado at Boulder, featured on Colorado Matters today, also provides some interesting insight into how galactic clusters, where MOND theory fails, differ from other cosmic phenomena. He notes that galactic clusters tend to appear at the intersections of hard to see gaseous macrofiliments of matter that seem to form the skeletal outline of the universe.