  We present a new approach to measure the shapes of galaxies, a fundamental
task in observational astronomy. This approach is based on the decomposition of
a galaxy image into a series of orthogonal basis functions, or `shapelets'. Our
choice of basis functions, namely the Gauss-Hermite series, has a number of
remarkable properties under distortions, convolutions and noise, which makes
them particularly well suited for astrophysical applications. In particular, we
describe how they can be used to measure the shear induced by weak
gravitational lensing, with the precision required for upcoming surveys. We
also show how shapelets can be used to reconstruct images from interferometric
observations. Other application of shapelets, such as image compression, PSF
deconvolution, de-projection and the study of galaxy morphology, are also
briefly discussed.
