Image diffusion partial differential equations have been applied to medical and non-medical images in applications such as denoising, sharpening, and interpolating missing data, with great success. In this talk, we will review two recent geometrical approaches -- Beltrami and Hypoelliptic -- that have been proposed for deriving image diffusion equations, and we will present theoretical and experimental results on a class of geometrically inspired diffusion equations that we have recently proposed. The diffusion equations we obtain are derived by changing the Riemannian metric on the space of images from L^2 to Sobolev, and lead to flows which could not be obtained under the standard L^2 metric (Joint work with J. Calder (Michigan) and A. Yezzi (Georgia Tech)).