Rollo led raids on Northern France in the 9th century. Rollo and his Viking men sailed up and down the river Seine terrorising the locals. The lands were rich in game and the rivers filled with fish; the Vikings liked what they saw. In 911 Rollo and the King of France struck a deal – King Charles III offered Rollo the lands he wanted if he followed 2 simple rules; 1, He defends the lands from other Viking raiders, and 2, he and his men convert to Christianity. Rollo agreed and Normandy was formed.
A legend has it that King Charles III asked Rollo to bow down and kiss his shoe once The Treaty of Saint Clair-sur-Epte was signed. Rollo refused and told one of his warriors to do it, the warrior keeled down, raised the Kings shoe, and pushed over, a sign the Normans never desired to be ruled over.
Now that Rollo had taken the lands with force, he realised he and his people needed to integrate with locals. They did this by marrying the local women and accepting their culture, ideology and even language. The borders of Normandy soon started to Expand and before long it had doubled in size.
The Normans continued to expand; in 1066 William the conqueror led a full invasion of England. In 1096 Williams son Robert was one of the leaders of the first Crusade. And in 1130 Roger II unified all of southern Italy to form The Kingdom of Sicily.
Most of the information and the picture are from BBC and the 3 part documentary The Normans, I recommend watching them. Link