In this paper, we present a general theory of adaptive mathematical morphology (AMM) in the Euclidean space. The proposed theory preserves the notion of a structuring element, which is crucial in the design of geometrical signal and image processing applications. Moreover, we demonstrate the theoretical and practical distinctions between adaptive and spatially-variant mathematical morphology. We provide examples of the use of AMM in various image processing applications, and show the power of the proposed framework in image denoising and segmentation.