One of the most intensively studied problems in image processing concerns how to detect edges in images. Edges are important since they mark the locations of discontinuities in depth, surface orientation, or reflectance, and their detection can facilitate a variety of applications including image segmentation and object recognition. Accurate detection of faint, low-contrast edges in noisy images is challenging. Optimal detection of such edges can potentially be achieved if we use filters that match the shapes, lengths, and orientations of the sought edges. This however requires search in the space of continuous curves. In this talk we explore the limits of detectability, taking into account the lengths of edges and their combinatorics. We further construct two efficient multi-level algorithms for edge detection. The first algorithm uses a family of rectangular filters of variable lengths and orientations. The second algorithm uses a family of curved filters constructed through a dynamic-programming-like procedure using a modified beamlet transform. We demonstrate the power of these algorithms in applications to both noisy and natural images, showing state-of-the-art results.