If you want to find dark matter, there’s one simple rule: you follow the mass. Indeed, if you look at the largest structures in the Universe — big galaxies, groups of galaxies, or even the most massive clusters — they all show the same thing: their internal motions are all too fast to be explained by the gravitation of the matter we know is there.
Inside individual spiral galaxies, their rotation speeds remain large, and in some cases even get larger and larger, as we move away from the galactic center. This cannot be explained by the sum total of all the different types of normal (atomic-based) matter we know exists: stars, gas, dust, plasma, even black holes.
Meanwhile, inside groups and clusters of galaxies, the speeds of the galaxies inside is also far too large to be explained by the normal matter. In all of these cases, if the only gravitation were due to the normal matter alone, these bound structures would fly apart, as their internal speeds are too great (greater than the escape velocity) for the mass due to all sources of protons, neutrons and electrons.
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