In this paper, after providing an appropriate coordinate system, we investigate the weighted distances on the Khalimsky grid. There are two types of natural neighborhood relations, and one semi-neighborhood is also defined. Weighted distances are defined for both cases, i.e., allowing or not the semi-neighborhood. We give formulae for computing the weighted distance of any point-pair on the Khalimsky grid in these cases. Digital disks based on the weighted distances are also investigated. In some cases, these disks may not be convex; moreover, they may contain holes. Sometimes, if semi-neighborhood is allowed, they are not connected, i.e., they contain islands. The conditions of concavities, holes and islands are characterized.