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). Alternate approaches have therefore been considered, one of them called the Modified Newtonian Dynamics (MOND) theory.