Signal compression is the process of finding a compact digital representation of a signal. Its aim is to reduce the bit-rate of a digital signal with or without loss of information about the signal. When compression is achieved with a loss of signal information, it is called lossy signal compression; otherwise it is called lossless signal compression. In the literature, the process of signal compression is also described by other names (such as signal coding, bandwidth compression, data compression, source coding, digital coding, etc.). In this paper, we will use the terms signal compression and coding alternately, both describing the same process. Currently, signal compression (or, coding) is very much a part of our everyday life. Its applications are primarily in transmission and storage. When you talk to your friend on a cellular phone, your speech is first converted into digital form and then compressed so that it can be transmitted over a limited-­bandwidth radio channel. Image and video clips are compressed prior to their storage on network servers. When you download an image through the Internet, you most likely receive a compressed image and your web browser decompresses it prior to its display on your computer. In this case, the compression process is reducing both the storage requirement on the net server and transmission time on the Internet. In the Signal Processing Laboratory at Griffith University, we are interested in the compression of telephone speech, wideband speech, audio, image and video signals. In this paper, we confine our scope to speech and image coding, and describe the research work done at our laboratory in the following three areas: 1) lossy speech coding, 2) lossy image coding, and 3) lossless image coding.