There are several dark matter problems, each on a different scale. The standard way to prove the existence of dark matter is to compare dynamical estimates of the mass of an object (galaxy, cluster) with an evaluation of its population contents. Historically, we can distinguish four distinct problems :
In each case there is more matter inferred dynamically than can be accounted for by known matter components. This mass discrepancy is usually attributed to additional (dark) matter, assuming that Newton's laws are valid. Only in Modified Newtonian Dynamics (MOND, cf. Milgrom 1983) is the discrepancy attributed to a modification of the force law at low densities. This theory has been worked out in detail only for the explanation of spiral galaxy rotation curves. The evidence for dark matter from extended rotation curves of spiral galaxies is considered the strongest, and this topic has thus spawned the most adhoc alternatives. Not only MOND, but also explanations based on magnetic forces have been considered. (cf. Battaner et al. 1992, but see Cuddeford & Binney 1993). For a review of alternative theories of gravity, see Sanders (1990).
In these lecture notes, I will only discuss certain aspects of the whole dark matter problem, with a distinct emphasis on dark matter in spirals, including the Milky Way. The lectures were given to an audience of scientists working mainly in celestial mechanics, hence the inclusion of a lot of introductory material.