Approaches to object information extraction from images should attempt to use the fact that images are fuzzy. In past image segmentation research, the notion of `hanging togetherness' of image elements specified by their fuzzy connectedness has been lacking. We present a theory of fuzzy objects for n-dimensional digital spaces based on a notion of fuzzy connectedness of image elements. Although our definitions lead to problems of enormous combinatorial complexity, the theoretical results allow us to reduce this dramatically. We demonstrate the utility of the theory and algorithms in image segmentation based on several practical examples.