In the beginning of the 1980s, the first observational evidence was reported that galaxies do not spin as expected according to then current theories. A galaxy is a collection of stars orbiting the bulge (the center of the galaxy). Since the orbit of stars is driven solely by the gravitational force, it was expected that stars at the edge would have an orbital period much larger than those near the bulge. For example, the Earth which is 150 million kilometers away from the Sun completes an orbit in one year, while it takes Saturn 30 years to do the same at a distance of 1.4 billion kilometers.
A similar behavior was expected from galaxies, even if the distribution of stars is more cloud-like. However, it became more and more apparent that stars at the edge of a galaxy move faster than expected.
Astronomers call this phenomenon the "flattening of galaxies' rotation curve". Basically, if one draws a curve describing the orbital speed of stars as a function of the distance from the center, he or she should obtain curve A in fig. 2 (dashed line). Data from telescopes give curve B (plain line). This curve, instead of decreasing asymptotically to zero, remains flat at large distance from the bulge. For comparison purpose, the same curve for the Solar system -- (properly scaled) -- is provided (curve C in fig. 2).
Reluctant to change Newton's law as well as Einstein's theory of relativity for galaxies only, scientists simply assumed that the rotation curve was flat because of the presence of a large amount of matter surrounding the visible part of galaxies. The new theory was that galaxies are embedded in a spherical halo of invisible, "dark" matter (see fig. 3). Since then, the search for dark matter has kept many astronomers busy, with mitigated success.
As time has passed, the hypothesis of dark matter halos encountered many problems, casting doubt on the validity of this model (although it is still the most widely accepted model). (It is surprising however that the dark matter theory has not been challenged yet on grounds of general scientific implausibility alone: as the mass of the dark matter is usually many times the normal mass of the galaxy, it would be bizarre to assume that the formation of the dark matter halo has been triggered by the normal matter (one might as well assume that the planets are responsible for the formation of the sun). One would have to conclude therefore that, on the contrary, the dark matter has led to the formation of the visible galaxy. This however would mean that all galaxies should show evidence for dark matter, which is clearly not the case).
Alternate approaches have therefore been considered, one of them called the Modified Newtonian Dynamics (MOND) theory.