Sparse representations of data play an increasingly important role in areas across applied mathematics, science and engineering. Over the past few years, there has been a rapidly increasing pressure to handle ever larger and higher dimensional data sets, with the challenge of providing representations of these data that are sparse and fast to compute. JPEG2000 is the latest compression scheme that makes use of the wavelet transform. However, the wavelet transform is known to be suboptimal when dealing with a certain class of images, e.g. images described as C 2 functions away from piecewise C 2 curves. In this talk, we develop a transform called the shearlet transform designed to handle such images optimally. We conclude with many examples indicating that shearlets are the latest competitor to beat for representing images sparsely.