The invention refers to an arrangement for positioning substrates, in particular for positioning wafers, within a device that is provided for exposure of the substrates and/or for measurement on the substrates by means of radiation under high-vacuum conditions. The following are provided according to the present invention: a retention system (4), displaceable on a linear guidance system (3), for receiving the substrate, the guidance direction of the linear guidance system (3) being oriented parallel or substantially parallel to the Y coordinate of an X, Y, Z spatial coordinate system; drives for limited modification of the inclination of the guidance direction relative to the Y coordinate; drives for limited rotation of the linear guidance system (3), including the retention system (4), about the guidance direction; and drives for parallel displacement of the linear guidance system (3), including the retention system (4), in the direction of the X coordinate, the Y coordinate, and/or the Z coordinate.