The discovery of dark matter should be assigned to the study of rotation curves of spiral galaxies. Indeed, a study that contained clear clues about dark halos was the measurement, in the late 1950s, by means of 21-cm radio observations, of the rotation curve of M31, the Andromeda galaxy. Therefore, this second part begins with a short introduction to galaxies. It starts with a description of how galaxies were discovered, because the topic illustrates a sort of paradigm for more recent discoveries in astrophysics, such as that of Gamma Ray Bursts. It continues with highlights on their morphology, their structure and kinematics, following a description of their typical scales, with special reference to the concept of dynamical time scale. All this should be considered classical information that is a prerequisite for any further discussion about the dynamics of galaxies. The subject of scales gives the opportunity to mention some modern developments about galaxies characterized by unusual properties that are currently studied at the frontier of galactic dynamics. The closing section of the chapter is devoted to introducing the virial theorem in its simplest form, for the discrete and the continuous case; this is an important tool at the the basis of many arguments about self-gravitating systems.

We start with some of the historical work on measuring distances to galaxies, leading to the Hubble (or Hubble–Lemaîe) law, a linear proportionality between recession velocity and and a galaxy’s distance, with a proportionality constant known as the Hubble constant. For more distant galaxies, it becomes increasingly difficult to detect and resolve even giant stars like Cepheid variables as individual objects, limiting their utility in testing the Hubble law to larger distances and redshifts. For much larger distances, an important alternative method is the Tully–Fisher relation.

The second variant of Milgrom’s theory (T1) derives the equations of motion from a Lagrangian formulation, allowing the theory to make predictions for a very general class of systems with arbitrary shapes. A number of confirmed, novel predictions follow from T1 including the “central surface density relation” (CSDR) and a prediction about the form of the vertical force law in the Milky Way. The theory also predicts a very low rate of mergers between galaxies, a prediction that may or may not be consistent with observations. However, the theory’s predictions about the kinematics of the largest bound structures in the universe, the galaxy clusters, appear to be incorrect.

The first variant of Milgrom’s theory (T0) consists simply of his three postulates from 1983. These postulates entail a number of novel predictions, predictions that have subsequently been confirmed by observational astrophysicists. The first is the “baryonic Tully–Fisher relation” (BTFR), a unique relation between the total mass of a galaxy and its asymptotic rotation speed. An even more surprising prediction is the “radial acceleration relation” (RAR), which states that the rotation speed anywhere in a disk galaxy is determined precisely by the observed distribution of matter – but not in the way that Newton’s laws would predict. According to Lakatos’s Methodology, these, and some other, successful novel predictions imply that Milgrom’s postulates constitute a progressive departure from the dark matter hypotheses of the standard cosmological model.