Currently (2015) there is a reconstruction on the bridge. It is not clear what is left of the original bridge, as the restored areas have already new surface.
Another popular place is the city's cathedral, which is visible from everywhere. Bishop's residence is located near the cathedral. From about 500 to 788 Regensburg was the capital of the Bavarian dukes Agilolfings. Then, until 1200 was the residence of the Bavarian bishops. And from 1200 to 1803 - the imperial city and has been directly subordinated to the emperor.
If you want to find the areas free of tourists, then come on embankments or to the islands on the Danube, they can be seen on the following map. There's a lot of greenery, the best scenery and few people.
Old Town Hall, now a museum, located on a cozy Town Hall Square. Here was announced the decay of the Holy Roman Empire.
There is also a large palace in the city, the former Abbey of St. Emmeram. but now it is home to billionaire Albert von Thurn and Taxis, one of the richest people in Germany, so the palace is closed to the public, but you can visit the Basilica. Representatives of the princely family of Thurn and Taxis live here since 1812.
There are regular activities on the territory of the palace, such as Christmas Fair.
Brewery, owned by family of Thurn and Taxis, is located next to the the residence.
Old Town is included in the list of the world heritage sites by UNESCO. Here you can see a lot of interesting old houses, especially near the Stone Bridge and the gates into the city.
Taking a walk through the old city, find such an unusual parts of the house. This is the remains of buildings from Roman times. At 79 AD the first fortress was built here.

Porta Praetoria - the northern gate of Castra Regina, have become part of the walls of the residence of the bishop of Regensburg ©Yuriy Buriak
Very interesting house-tower with different windows on each floor
Cozy yard
Guide to the city with a map and a description (16 Mb).
If you are traveling by car, we recommend Hotel Hotel Luis Stadl (booking.com). There are parking problems in the city, so you need hotel with good parking and only 10 minutes walk to the center.
Getting there: By train, use bahn.de. Travel time at the regional train from Munich 1:30, from Nuremberg 1:04.