Men from the north (North-man) went up the Seine river and settled in the region in the 10th century. Rollo the invader was given a Ducthy byThe King of France granted in exchange for his vassalage. The Normans influence extended to the west. The old borders of normandy have remained pretty much the same since then.
In 1066, the Duke of Normandy, William the Conqueror invaded England at the battle of Hastings and became king of Englang. England was French for 80-years!
At the same time the Normans invaded the Kingdom of Antioch and southern Italy and Sicily. Their strategy was assimilation. Therefore in the 13th century there is no longer "Normans"-strictly speaking- with their specificities. Being Normand remains more as a identity than a reality.
Normandy became british in 1144 with the Plantagenets.
Philippe Auguste takes Normandy back in 1204 making it French again.
It goes back as to England in 1346 during the 100 years war and the invasion of France by Edward III of England.
This back and forth exchange is completed in 1469. With the end of the conflict, Normandy is definitely French.
Normandy was 3 times British and 4 times French. It is a land of conflict and encounter between these two Peoples. It has developped it own identity. An identity that is full of British influence and not quite French
The strong bounds between France and England manifested again in June 1944 with the Allied landing, strongly affecting the region.