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Photo: Richard Schubert, CC BY-SA 3.0 via Wikimedia Commons
About 6 km north of Tübingen, right in the Schönbuch forest reservation, is the old cistercian monastery (founded 1187 AD) of with a small village next to it, especially nice to visit at night when part of the monastery is illuminated.
Bebenhausen Abbey is a former Cistercian monastery complex located in Bebenhausen, Baden-Württemberg, Germany. The complex is also the location of Bebenhausen Palace, a hunting retreat created and maintained by two Kings of Württemberg. The complex was named a historic monument in 1974.
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A small modern art museum on one of the hills.
Stadtmuseum
The old city centre full of crooked half-timbered houses, small alleys, cobblestone roadwork and some nice old churches.
Hölderlinturm
The river Neckar next to the city centre, on which the students go punting (similar to what they do in Oxford or Cambridge in summer time, occasionally with a barbecue and a crate of beer on board.
The Schloss, or castle, can be a fun place to wander around. Look at the intricately carved gate at the entrance. Skip most of the centre courtyard and head up the stairs on the back side of the courtyard. This leads to a beautiful garden/courtyard, with some beautiful views from atop the wall to the left. Alternatively, go through the tunnel on the backside of the main courtyard that will take you through the backside of the castle, a much more medieval and beautiful part of the Schloss. At certain times, there are archers shooting in the eastern moat (the "Hasengraben") of the castle. From back here there is a path that will take you along the wall of the castle and through the woods, ending up on Haaggasse very near the Marktplatz.