In signal and image processing, we often use transforms in order to simplify operations or to enable better treatment to the given data. A recent trend in these fields is the use of over complete linear transforms that lead to a sparse description of signals. This new breed of methods is more difficult to use, often requiring more computations. Still, they are much more effective in applications such as signal compression and inverse problems. In fact, much of the success attributed to the wavelet transform in recent years, is directly related to the above-mentioned trend. In this talk we will present a survey of this recent path of research, and its main results. We will discuss both the theoretic and the application sides to this field. No previous knowledge is assumed (… just common sense, and little bit of linear algebra).