We compare a range of powerful compression methods - fractal, wavelets, pyramidal median, JPEG - with compression tools dedicated to astronomy such as HCOMPRESS, FITSPRESS and Mathematical Morphology, and apply these to astronomical images. Quality is quantified from visual appearance, and from photometric and astrometric measurements. Computational requirements of each method are discussed. We also review the implications of Web-based storage and transmission, stressing what we term progressive vision. In summary, no method is perfect, but the PMT method is the best compromise for general astronomical images, combining acceptable photometric and positional precision with good compression capabilities. JPEG is still an excellent method for compression factors less than 40 and has the advantage of being very widely available.