Image curves are one of the choices for representing interest points which also provide discriminative information about images. Boundary of regions and contour of shapes are real-time instances of image curves. In this thesis, we propose two approaches for keypoint detection and description on image curves. To extract keypoints on image curves, we compute the extrema curvature of region boundaries. This mechanism improves repeatability of keypoints on 3D data. For the description of image curves, shape contours are used. This is similar to approaches that describe the features based on shapes and image gradients. Unlike these approaches, we combine spatial and directional information of tangent directions to extract a feature vector that leads to improved matching and recognition on several standard computer vision tasks such as character and object recognition.