Astronomy is arguably the oldest scientific discipline. Precise measurements of the motion of celestial bodies date back to the ancient Babylonians, Chinese, and indigenous peoples outside Eurasia. Starting in the 19^{th} century, systematic applications of physical principles to the formation and dynamics of stars marked the birth of astrophysics as a subfield of physics. Present-day astrophysics employs an array of theoretical and observational tools to construct sophisticated and predictive models of the origin, evolution, and death of stars.

While stars can be largely described within Newtonian physics, some of their most interesting properties, such as bounds on their radius-mass ratio, their potential collapse into a black hole, or effects of viscosity in gravitational waves emitted by mergers of neutron stars, can only be studied via applications of general relativity. Moreover, as a matter of principle, we ought to be able to fully understand stars as general-relativistic phenomena. The mathematical treatment of stars within general relativity, however, has lagged. Little progress has been made on this front since the discovery of the Tolman-Oppenheimer-Volkoff (TOV) equations and the Oppenheimer-Snyder solution in the late ‘30s. The former describes a static (i.e., time independent), perfectly spherically symmetric star, whilst the latter describes the collapse of a perfectly spherically symmetrical star with no pressure into a black hole. Despite being landmark results in general relativity, both situations are highly idealized. Inferences about the general properties of general-relativistic stars from them is, therefore, a priori unjustified.

In this talk, I will discuss the problem of formulating a sound mathematical theory of general-relativistic star evolution. After setting up the problem, I will explain its main challenges, but also how a great deal of rich physics and mathematics is involved in its study. A fundamental difficulty involves understanding the mathematics of the fluid-vacuum interface which separates the body of the star from vacuum. This interface displays a singular behavior which is not amenable to current mathematical techniques. This difficulty, however, can be circumvented if we consider stars that are spherically symmetric but not static. This corresponds to a dynamic (i.e., time-dependent) generalization of the TOV equations. I will conclude with some possible directions of future research, including the treatment of general-relativistic viscous star models. This is joint work with J. Speck.