Particle-like objects have a very long and broad history in science, arising long before Newton’s corpuscles of light, and spanning the range from fundamental to astronomical. In the mid-1950s, John Wheeler sought to construct stable, particle-like solutions from only the smooth, classical fields of electromagnetism coupled to general relativity [220, 182]. Such solutions would represent something of a “gravitational atom”, but the solutions Wheeler found, which he called geons, were unstable. However, in the following decade, Kaup replaced electromagnetism with a complex scalar field , and found Klein–Gordon geons that, in all their guises, have become well-known as today’s boson stars (see Section II of  for a discussion of the naming history of boson stars).
As compact, stationary configurations of scalar field bound by gravity, boson stars are called upon to fill a number of different roles. Most obviously, could such solutions actually represent astrophysical objects, either observed directly or indirectly through its gravity? Instead, if constructed larger than a galaxy, could a boson star serve as the dark matter halo that explains the flat rotation curve observed for most galaxies?
The equations describing boson stars are relatively simple, and so even if they do not exist in nature, they still serve as a simple and important model for compact objects, ranging from particles to stars and galaxies. In all these cases, boson stars represent a balance between the dispersive nature of the scalar field and the attraction of gravity holding it together.
This review is organized as follows. The rest of this section describes some general features about boson stars. The system of equations describing the evolution of the scalar field and gravity (i.e., the Einstein–Klein–Gordon equations) are presented in Section 2. These equations are restricted to the spherical symmetric case (with a harmonic ansatz for the complex scalar field and a simple massive potential) to obtain a boson-star family of solutions. To accommodate all their possible uses, a large variety of boson-star types have come into existence, many of which are described in more detail in Section 3. For example, one can vary the form of the scalar field potential to achieve a large range of masses and compactness than with just a mass term in the potential. Certain types of potential admit soliton-like solutions even in the absence of gravity, leading to so-called Q-stars. One can adopt Newtonian gravity instead of general relativity, or construct solutions from a real scalar field instead of a complex one. It is also possible to find solutions coupled to an electromagnetic field or a perfect fluid, leading respectively to charged boson stars and fermion-boson stars. Rotating boson stars are found to have an angular momentum which is not arbitrary, but instead quantized. Multi-state boson stars with more than one complex scalar field are also considered.
We discuss the dynamics of boson stars in Section 4. Arguably, the most important property of boson-star dynamics concerns their stability. A stability analysis of the solutions can be performed either by studying linear perturbations, catastrophe theory, or numerical non-linear evolutions. The latter option allows for the study of the final state of perturbed stars. Possible endstates include dispersion to infinity of the scalar field, migration from unstable to stable configurations, and collapse to a black hole. There is also the question of formation of boson stars. Full numerical evolutions in 3D allow for the merger of binary boson stars, which display a large range of different behaviors as well producing distinct gravitational-wave signatures.
Finally, we review the impact of boson stars in astronomy in Section 5 (as an astrophysical object, black hole mimickers and origin of dark matter) and in mathematics in Section 6 (studies of critical behavior, the Hoop conjecture and higher dimensions). We conclude with some remarks and future directions.
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