Her relationship with the Nazis is complicated, and is only relatively recently being studied and interrogated more deeply, but there are broadly two schools of thought on the matter:

• She was either a pure opportunist or, at worst, an anti-Semite and Nazi sympathiser. She had a longstanding affair with an Abwehr (German military intelligence) officer, Hans Gunther von Dincklage, enabling her to live a comfortable life during the occupation of France — the pejorative term 'horizontal collaborator' is commonly used here.

or:

• All of the above is true — and additionally, that in late 1941 she was recruited by either Dincklage, another Abwehr agent named Hermann Niebuhr or SS intelligence chief Walter Schellenberg as an agent of the Abwehr. This school of thought claims that Chanel led a somewhat inglorious career as a German asset, including an abortive plan (Operation Modellhut) to travel to Madrid and make clandestine contact with the British government — leveraging her relations with several British notables, including her former lover, the Duke of Westminster — in order to broker an end to the war.

Allegations about Chanel's wartime activities have persisted since the 1940s, and she was interrogated by the French authorities after the war, but no action was taken — allegedly thanks to some wily manoeuvring and pulling of strings on her part.

The major (recent) source for the second claim is a 2011 book by American journalist Hal Vaughan, Sleeping with the Enemy: Coco Chanel's Secret War. The book is not without its critics, but he presents some pretty compelling evidence, including documents from the archives of both French and German intelligence services.

Chanel consistently denied that she worked for the Germans; her defenders generally claim that the fact she was given an Abwehr agent number and assigned a codename (WESTMINSTER) is inconclusive, and that many people were registered in this manner without their knowledge or any action on their part. That's Vaughan's cornerstone evidence, and I agree (to a point) that it's not ironclad — but there does seem to be a pretty strong case to be made that Chanel existed in close proximity to several German intelligence officers, and that her relationship with the occupation authorities went far deeper and was far more involved than she ever admitted.