Laurussia formed when 3 continents Baltica, North America, and Avalonia collided with each other during the Silurian. Baltica and Avalonia collided with each other first in the Shelveian Phase of the Caledonian orogeny. Then Avalonia and Baltica collided with North America in the Acadian Phase of the Caledonian orogeny. This drift of Avalonia closed the Iapetus Ocean and opened the Rheic Ocean behind it.
Laurussia became part of Pangaea in the Permian. In the Jurassic, when Pangaea rifted into two supercontinents, Gondwana and Laurasia, Laurussia became part of Laurasia. The Atlantic ocean opened up in the middle of Laurussia, causing it to split into North America and Europe.