Within the last 20 years, gravitational lensing has changed from being considered a geometric curiosity to a helpful and in some ways unique tool of modern astrophysics. Although the deflection of light at the solar limb was very successfully hailed as the first experiment to confirm a prediction of Einstein’s theory of General Relativity in 1919, it took more than half a century to establish this phenomenon observationally in some other environment. By now almost a dozen different realizations of lensing are known and observed, and surely more will show up.
Gravitational lensing – the attraction of light by matter – displays a number of attractive features as an academic discipline. Its principles are very easy to understand and to explain due to its being a geometrical effect. Its ability to produce optical illusions is fascinating to scientists and laypeople alike. And – most importantly of course – its usefulness for a number of astrophysical problems makes it an attractive tool in many branches of astronomy. All three aspects will be considered below.
In its almost two decades of existence as an observational branch of astrophysics, the field of gravitational lensing has been continuously growing. Every few years a new realisation of the phenomenon is discovered. Multiple quasars, giant luminous arcs, quasar microlensing, Einstein rings, galactic microlensing, weak lensing, galaxy-galaxy lensing open up very different regimes for the gravitational telescope. This trend is reflected in the growing number of people working in the field. In Figure 1 the number of publications in scientific journals that deal with gravitational lensing is plotted over time. It is obvious that lensing is booming as an area of investigation.
Although there had been a slight sense of disappointment in the astronomical community a few years ago because lensing had not yet solved all the big problems of astrophysics (e.g. determination of the Hubble constant; nature of dark matter; physics/size of quasars), this feeling has apparently reversed. With its many applications and quantitative results, lensing has started to fulfill its astrophysical promises.
We shall start with a brief look back in time and mention some historic aspects of light deflection and lensing in Section 2. We then attempt to explain the basic features of gravitational lensing quantitatively, deriving some of the relevant equations (Section 3). A whole variety of lensing observations and phenomena which curved space-time provides for us is presented in Section 4, for example, multiple versions of quasars, gigantically distorted images of galaxies, and highly magnified stars. Additionally, we explain and discuss the astrophysical applications of lensing which show the use of this tool. This section will be the most detailed one. Finally, in the concluding Section 5 we try to extrapolate and speculate about the future development of the field.
By design, this review can only touch upon the issues relevant in the astrophysical field of gravitational lensing. This article should serve as a guide and general introduction and provide a number of useful links and references. It is entirely impossible to be complete in any sense. So the selection of topics and literature necessarily is subjective. Since the idea of the “Living Reviews” is to be regularly updated, I ask all authors whose work I may not have represented properly to contact me so that this can be corrected in the next version of this article.
The textbook by Schneider, Ehlers, and Falco  contains the most comprehensive presentation of gravitational lensing. A new edition is underway. The book by Bliokh and Minakov  on gravitational lensing is still only available in Russian. A new book currently in press by Petters, Levine, and Wambsganss  treats mainly the mathematical aspects of lensing, in particular its applications to singularity theory.
The contributions to the most important conferences on gravitational lensing in the last few years have all been published: Swings  edited the Proceedings on the first conference on lensing in Liège in 1983. Moran et al.  are the editors of the MIT workshop on lensing in 1988. Also see, Mellier et al.  of the Toulouse conference in 1989; Kayser et al.  of the Hamburg meeting in 1991; Surdej et al.  of the Liège conference in 1993; and Kochanek and Hewitt  of the IAU Symposium 173 in Melbourne in 1995. Online proceedings of a few smaller and more recent meetings also exist. See: Jackson  of the Jodrell Bank Meeting “Golden Lenses” in 1997.
A number of excellent reviews on gravitational lensing also exist. Blandford and Kochanek  give a nice introduction on the theory of lensing. The optical aspects of lensing are derived elegantly in . The presentation of Blandford and Narayan  emphasizes in particular the cosmological applications of gravitational lensing. The review by Refsdal and Surdej  contains a section on optical model lenses that simulate the lensing action of certain astrophysical objects. A recent review article by Narayan and Bartelmann  summarizes in a very nice and easy-to-understand way the basics and the latest in the gravitational lens business. In the sections below, some more specific review articles will be mentioned.
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