Unseen dark matter has been invoked several times to solve problems in astrophysics and cosmology. Historically the most significant problem has been the rotation curves of galaxies, particularly spiral galaxies. Using the Doppler Effect the speeds of the stars and gases in the disk regions of spiral galaxies can be measured. See Figure 1.
By now hundreds of thousands of galaxies have been measured this way. What is observed is that the speeds of the stars, and the gases beyond where the stars are observed, are much greater than it would appear Newtonian physics allows for.
As a result it has been suggested that there is an invisible halo of cold non-interacting matter. This putative invisible halo has the needed gravitational effect on the stars and gases but it cannot be seen, hence it is called dark matter. Dark matter is alleged not to be normal atomic matter, made from protons and neutrons (which are known as baryons), but some sort of slowly moving (cold) exotic non-baryonic matter. Weakly Interacting Massive Particles (WIMPs) were suggested.