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Christkindlmarkt
At Christmas, the historic inner city transforms into a seasonal event. The heart is traditional Christkindlmarkt (Christmas Market), with wooden stalls selling food, Christmas gimmicks, Glühwein (lit. "glowing wine", warm seasoned wine) and other things, a merry-go-round and a story-teller. Another highlight is the outdoor ice skating rink in front at Paradeplatz.
Literally translated "The Duke's Box" this old castle built in 1255 is the oldest secular building in the city. Today it hosts the city library.
An art museum.
Bayrisches Armeemuseum
The museum is separated into two locations. The Neues Schloss, a former duke's residence and part of the city wall, hosts a collection of military artefacts from different eras as well as a bunch of replicas. The Reduit Tilly, a part of the former state fortress in Klenzepark hosts a World War I exhibition. The latter is a very interesting and well structured museum, with exhibits and documents from the era and some short films which summarise the history. The Army Museum is being renovated in 2019, so some parts may be closed. Many of the captions are in German and they have something of a pro-Bavarian and politically conservative slant but the original exhibits are interesting even if you don't subscribe to their view of history.
South of the inner city, on the right side of the river Danube, lies the Reduit Tilly, the core building of the classical Brückenkopf (Bridge Head) fortress. You can visit the World War I exhibition of the Bavarian Army Museum inside or take a walk around the surrounding park area (Klenzepark).
Altes Rathaus
Dating back to the 14th century, this building was created by fusing different buildings into one in 1882. It is the seat of the mayor and - what might be of interest to a traveller even more - the city's tourist information.