If you like to know a little about the history of the places you travel to or live in, then this is a great city museum, well worth a couple hours of your time. Having visited the city museum, I was able to appreciate my second trip to the city so much more than my first go around. For those who are English monolingual, note that they provide FREE audio tour handsets. A few recordings here and there didn't play, but that didn't detract too much from the experience.
The museum isn't really arranged in chronological order. When you enter, you'll first see some displays regarding some of the hallmarks of Cologne - kölsch beer, the puppet shows, Karneval, and some pop culture. Then, you'll come to some WWII displays that offer very rewarding insight into Cologne's experience during the Second World War. From there, the World War I era artifacts and World War II artifacts get a little mixed, and then the exhibits jump straight to the early Middle Ages.
Arrangement aside, I found the exhibits to be interesting and informative. They trace Cologne's development into a commercial and world power, its role in power struggles between church and secular governments, its myths and legends, Jewish community, and of course, its role in modern topics, such as evolving family/social roles and the impacts of the political upheavals and philosophies of the 19th and 20th centuries.
I would recommend getting the MuseumsCard. It gets you into all public museums in Cologne for two days, and also gives you free public transit for the first day. The MuseumsCard costs €15 for a single, or €28 for a family card (two adults and two children). read more