Any non-uniform matter distribution between our observing point and distant light sources affects the measurable properties of the sources in two different ways: The angular size of extended objects is changed and the apparent brightness of a source is affected, as was first formulated in 1967 by Gunn [71, 72].
A weak lensing effect can be a small deformation of the shape of a cosmic object, or a small modification of its brightness, or a small change of its position. In general the latter cannot be observed, since we have no way of knowing the unaffected position9.
The first two effects – slight shape deformation or small change in brightness – in general cannot be determined for an individual image. Only when averaging over a whole ensemble of images it is possible to measure the shape distortion, since the weak lensing (due to mass distributions of large angular size) acts as the coherent deformation of the shapes of extended background sources.
The effect on the apparent brightness of sources shows that weak lensing can be both a blessing and a curse for astronomers: The statistical incoherent lens-induced change of the apparent brightness of (widely separated) “standard candles” – like type Ia supernovae – affects the accuracy of the determination of cosmological parameters [62, 88, 196].
The idea to use the weak distortion and tangential alignment of faint background galaxies to map the mass distribution of foreground galaxies and clusters has been floating around for a long time. The first attempts go back to the years 1978/79, when Tyson and his group tried to measure the positions and orientations of the then newly discovered faint blue galaxies, which were suspected to be at large distances. Due to the not quite adequate techniques at the time (photographic plates), these efforts ended unsuccessfully [183, 189]. Even with the advent of the new technology of CCD cameras, it was not immediately possible to detect weak lensing, since the pixel size originally was relatively large (of order an arcsecond). Only with smaller CCD pixels, improved seeing conditions at the telecope sites and improved image quality of the telescope optics the weak lensing effect could ultimately be measured.
Weak lensing is one of the two sub-disciplines within the field of gravitational lensing with the highest rate of growth in the last couple of years (along with galactic microlensing). There are a number of reasons for that:
Now we will briefly summarize the technique of how to use the weak lensing distortion in order to get the mass distribution of the underlying matter.
The first real detection of a coherent weak lensing signal of distorted background galaxies was measured in 1990 around the galaxy clusters Abell 1689 and CL1409+52 . It was shown that the orientation of background galaxies – the angle of the semi-major axes of the elliptical isophotes relative to the center of the cluster – was more likely to be tangentially oriented relative to the cluster than radially. For an unaffected population of background galaxies one would expect no preferential direction. This analysis is based on the assumption that the major axes of the background galaxies are intrinsically randomly oriented.
With the elegant and powerful method developed by Kaiser and Squires  the weak lensing signal can be used to quantitatively reconstruct the surface mass distribution of the cluster. This method relies on the fact that the convergence and the two components of the shear , are linear combinations of the second derivative of the effective lensing potential (cf. Equations (33, 34, 35)). After Fourier transforming the expressions for the convergence and the shear one obtains linear relations between the transformed components , , . Solving for and inverse Fourier transforming gives an estimate for the convergence (details can be found in [87, 86, 126] or ).
The original Kaiser-Squires method was improved/modified/extended/generalized by various authors subsequently. In particular the constraining fact that observational data are available only in a relatively small, finite area was implemented. Maximum likelihood techniques, non-linear reconstructions as well as methods using the amplification effect rather than the distortion effect complement each other. Various variants of the mass reconstruction technique have been successfully applied to more than a dozen rich clusters by now.
Descriptions of various techniques and applications for the cluster mass reconstruction can be found in, e.g., [1, 15, 17, 19, 29, 33, 73, 85, 127, 168, 171, 172, 174, 206]. In Figure 19 a recent example for the reconstructed mass distribution of galaxy cluster CL1358+62 is shown .
An upcoming comprehensive review on weak lensing by Schneider & Bartelmann  treats both theory and applications of weak lensing in great depths.
© Max Planck Society and the author(s)