For fancy occasions Roman men always continued to wear their wool robes over their tunics. They called these wool robes togas, and there were a lot of rules about how exactly a man should wear his toga, and who could have a stripe on his toga, and so forth, which helped to show who was rich and powerful and who was poor.
Women often wore more than one linen tunic, and a wool scarf or veil over their tunics, which they could pull over their heads if it was raining or cold.
When the Roman Empire controlled provinces all over Europe and the Mediterranean, the people in these different places continued to dress in their local ways, just putting on tunics and togas for special occasions. In Europe, in France and England and Germany, many people wore warm hooded cloaks, like the ones shown here.