This paper presents a review, analysis, and comparison of numerical methods implementing the curvature motion and the affine curvature motion for two-dimensional (2D) images, shapes, and curves. These curvature scale spaces allow, in principle, one to compute an accurate multiscale curvature in digital images. The fastest and most invariant of them can be used in a complete image processing chain. This numerical chain simulates the accurate subpixel evolution of an image by mean curvature motion or by affine invariant curvature motion. To do so, it lets all the level lines of the image evolve by curvature shortening (of affine shortening), computes the image curvature directly on the smoothed level lines, and reconstructs the evolved image and its curvatures in an intrinsic, grid independent representation. The paper describes a careful implementation of this chain and analyzes its effects on many examples. The microscopic visualization of an image curvature map reveals after processing many image details. This image process improves graphic images and gets rid of compression and aliasing effects. It also gives an accurate tool to explore the validity of Attneave's and Julesz's theories on shape perception and texture discrimination. The "curvature microscope'' runs online for any image at http://www.ipol.im/pub/algo/cmmm_image_curvature_microscope/.