The Kölner Dom, or Cologne Cathedral, dates back to the 13th century and is Cologne's best-known landmark. It's a fascinating historical building - about as Gothic as Gothic can be, and close examination of the intricate internal carvings reveals even the odd pagan symbol.
For those in good health, I recommend climbing the stairs of the spire. Sure, it's a particularly touristy thing to do, but climbing the spire is to Cologne as climbing the Arc de Triomphe is to Paris - it gives you a view of the city from a central vantage point, atop an amazing structure.
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