During their dominion the Normans built a great deal of castles in order to consolidate the king's power and to protect/ defend the territory. Even the central part of the Tower of London was built during the Norman age, more precisely during William’s reign, immediately after the conquest (He brought directly from Normandy the stones with which he built the tower).
A great deal of cathedrals date/ trace back to Norman age.
At first when they started building their early monuments they followed/ imitated/ built them in the Romanesque style which is a bit different from the Gothic one that represents the final evolution of Norman art. Norman art spread not only in England but even among Europe. Actually their early monuments are characterized by precise features like the presence of rounded arches and solidity.
Later on Gothic style started prevailing; it had completely different features: pointed arches, soaring spires and soaring heights. The presence of both styles can be observed/it’s possible to be seen in some monuments for example the Tower of London or Notre Dame because over the years different and successive bishops continuously added new elements to the original structures and rebuilt/ modified the preexisting structures. In many cathedrals these two styles mixed/ fused/ mingled for this reason.