The digital Mathematical Morphology and the Distance Transform (DT) have many points of intersection. The DT combines numerical features and objects shapes. The properties of digital distance functions (metrics, asymmetries and quasi-metrics) and DT based on these functions are studied. Some extensions of the transform and the interpretation of the grey scale DT by application of the binary DT in the n-dimensional digital space are given. The morphological erosion and dilation may be performed by the DT for binary and grey scale images.