Shapelets are a complete, orthonormal set of 2D basis functions constructed from Laguerre or Hermite polynomials weighted by a Gaussian. A linear combination of these functions can be used to model any image, in a similar way to Fourier or wavelet synthesis. The shapelet decomposition is particularly efficient for images localized in space, and provide a high level of compression for individual galaxies in astronomical data. The basis has many elegant mathematical properties that make it convenient for image analysis and processing.