Named after the spa town in Germany, where Prince Albert (later King Edward VII) of UK punched a dent in a hard felt hat and created this style in 1860. The stiffened, near-formal Homburg has a wide ribbon and a curled brim, with a bound edge. It was made famous again by Winston Churchill, Anthony Eden, Al Pacino in The Godfather and fictional detective Hercule Poirot. Some are made to be rigid, while others are as soft as Fedoras.
You have to be quite brave to wear a Homburg for the first time. It is a hat which makes a strong statement, as if a Fedora had fallen in love with a Bowler. But you quickly adapt to the outrageous curves and the sheer charisma of the Homburg.