Most parts right of the Elbe are located within the Nationalpark Sächsische Schweiz with all the rules you'd expect associated with this, but climbing is generally allowed. Left of the Elbe, the area is mostly covered by a lesser degree protection area (Landschaftsschutzgebiet), also with a well marked network of hiking trails. Some few summits are located on private property, see the guidebook. During spring and early summer, selected summits may be closed because of nesting falcons or owls - check at http://dav-felsinfo.de .
Sächsische Kletterregeln
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