Friday, October 1, 1943. The Germans depart, and destroy, Naples.

The U.S. 5th Army entered Naples.  The Germans burned the University of Naples and the Teatro di San Carlo on the way out.

German wartime poster attempting to recruit Italians to an SS formation.

The German appeal to the Italians didn't seem to be well calculated.

Hitler ordered Kesselring to hold a line south of Rome.

As a total aside, it was this date in 1943 that the German police and the Waffen SS adopted the M43 field cap, the famous billed cap that was patterned on the Bergmütze that had long been worn by German alpine troops. A similar hat had long been in Finnish use, nicknamed the "blood ladle".  German and Austrian alpine troops still wear a variant of it, as do many German hunters.

A very practical cap, the design had spread from mountain troops to the regular German Heer in a variant for North Africa, early versions of which had the ear flaps fixed in place, as some variants still do (including a variant now used as the Ukrainian field cap), but which the later versions allowed for them to be folded down, as with the Bergmütze.  The Afrika Korps version had a longer bill, which was retained for the M43.  When adopted by the Heer, it replaced the flat cap (garrison cap) which had been adopted in the 1930s.  The flat cap is a fairly useless cap, and the Bergmütze was a very practical one.  

As Waffen SS mountain units had already adopted the Bergmütze, they were allowed to fix the Edelweiss cap badge to their caps on this day, that being a symbol previously used only by the Heer.

Sarah Sundin noted the liberation of Naples and another item:

Today in World War II History—October 1, 1943: In Italy, US Fifth Army and British X Corps enter Naples. US Third War Loan Drive ends, raising $19 billion (quota $15 billion).

W. Averll Harriman was named Ambassador to the USSR