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In the early 1970s, astronomer Vera Rubin was working on undergraduate studies with her students. Over the course of many year, Rubin and her students plotted to rotational rates of tens of galaxies. A mystery started to emerge. Since the visible matter of the galaxy clearly looked higher (it was, after all, brighter and had more larger stars), it would be expected that the farthest reaches of the galaxies would spin slower to conserve angular momentum. Unfortunately, this was not the case, as the rotation rate of the galaxy remained pretty much constant throughout the galaxy. In fact, the hundreds of possible equations describing the "drop"-off rate of rotation all are that it seems to be logarithmic. There seemed to be more matter on the outside of galaxies.
Rubin's groundbreaking answer to this was that there had to be more matter than what being observed. A ring of "dark matter" surrounding galaxies would fit the predictions perfectly. Later research would reveal that this idea had already been proposed, although previous astronomers were right for the wrong reasons (see Oort and Zwicky and Slipher). It is now widely accepted that this is indeed the correct solution to the problem, with more observation supporting the existence of some invisible dark matter. The original problem has come to be known as the "Galaxy Rotation Problem", and it marks the start of the astronomical revolution that kickstarted the fields of cosmology and mathematical astrophysics.
This is not to say there aren't competing explanations for this phenomenon. The most most significant contenders are based on the contention that we do not know how gravity works on large scales, and that the observed results reveal the larger patterns of Newtonian gravity, the central idea of the framework of MOND (Modified Newtonian Dynamics). Since this eschews relativity, it is not mainstream, but the variant scalar-tensor-vector gravity (STVG) is at least accepted, and has gained a small following. It still isn't widely accepted.