In computer vision and image processing, edge detection concerns the localization of significant variations of the grey level image and the identification of the physical phenomena that originated them. It is difficult to design a general edge detection algorithm which performs well in many contexts and captures the requirements of subsequent processing stages. Consequently, over the history of digital image processing, a variety of edge detectors have been devised which differ in their mathematical and algorithmic properties. In this paper, we briefly describe some existing edge detectors. We investigate the wavelet transform with direction which is related to the Canny and Marr-Hildreth's edge detectors. This wavelet transform has a remarkable feature that plays an important role in image processing applications. We discuss its applications to edge and target detections. The direction effect enables the wavelets to analyze the directional features of images. Algorithms and experiments for edge and target detections have been developed based on the theory. And, very promising results have been shown in image processing. © World Scientific Publishing Company.