1. J. H. Oort: Astronomers have come to tust what is known as the mass to light ratio, M/L, that does a good job telling you what the mass of luminous matter should be based off of the luminosity of that matter. This relation is normalized such that for the sun, M/L = 1. In the 1930, Oort found that the stars moving on the galactic plane were moving faster than the galaxy's escape velocity! He knew what the mass of the visible matter should be from M/L and discovered stars in the galatic plane are moving too fast to be bound by that much mass. He postulated more mass must be present in the galaxy than can be attributed to the visible matter.
2. F. Zwicky: Zwicky studied the Coma Cluster and found that the stars had much more kinetic energy than they should from the viral theorem, KE
4. D. Walsh et al: in 1979, D. Walsh et al. were among the first to detect gravitational lensing. They watched how the light was bent by certain distant galaxies. The problem was that the galaxies had to have more mass to bend the light as profoundly as it did than come from the M/L relation. Dark matter could explain this discrepancy.
5. Microlensing: Several MACHOS studies went into effect and all came to the same conclusion: the missing matter could not be attributed to brown dwarfs, neutron stars, black holes, planets or other "dark" objects made of matter that we are familiar with. This extra mass had to be coming from some exotic type matter thus far unknown.
6. BBN: Big Bang Nucleosynthesis is one of the great achievements of modern cosmology. It turns out, the Deuterium to Hydrogen ratio (D/H) is heavily influenced by the overall density of baryons in the universe. Using D/H, one finds that the amount of baryons in the universe is much smaller than the total baryonic matter. The rest must be coming from some extra dark matter.
7. The CMB: The power spectrum taken from the Cosmic Microwave background is highly sensitive to the amount of baryonic matter in the universe. See the plot above. As the amount of baryons, \Omega_b, changes, so does the power spectrum... by a lot! The red error bars show the measured value. As can be seen, baryonic matter only makes up 4.6% of the universe. From the same power spectrum on finds that the total matter in the universe is more like 25% of the universe indicating that the vast majority of the matter is matter we don't understand.
8: N-Body Simulations and SDSS: Numerical simulations of large structure formation have been performed. Only those that include dark matter give results that match what we observe from large structure surveys such as the Sloan Digital Sky Survey.
9. The Bullet Cluster: "Smoking gun" evidence for dark matter, as some would say, came from a recent experiment involving the Bullet Cluster. The Bullet Cluster recently collided with a larger galaxy. In such a collision, dark matter should just pass through without interacting and the visible matter heated up giving a tremendous amount of X-Ray emissions. It was clear that the matter causing the majority of the lensing was not centered in the same spots as the luminous matter. This showed convincingly that the amount of baryonic matter in galaxies is not as large as the amount of dark matter. Furthermore, in 2007 another team confirmed a ring-like structure of dark matter was found after the collision of two massive galaxies.
10. Penny et al: In 2009, Penny et al. found that a significant amount of dark matter would be needed to hold certain galaxies together that were experiencing a significant amount of tidal forces. These galaxies were surprisingly stable given how little luminous mass they had.
As you can tell. This is a great article and I recommend everyone read it.
All images taken from the article cited.
Katherine Garrett, & Gintaras Duda (2010). Dark Matter: A Primer Eprint arXiv: 1006.2483v1