Yes, they did actually. The one common trait each of the conquests shared that I can currently source properly is that they all had the Pope's backing, at least at first in reference to Italy. Suffice to say they were also ambitious, opportunistic, fortune-seeking, land-hungry, and ferocious in battle. They seemingly had an adequate supply of landless (at first) yet capable knights and mercenaries to leave their lands in Normandy to seek wealth for themselves.

England - William II, Duke of Normandy had Pope Alexander II's permission to invade the Catholic Kingdom of England. The short answer is the Pope recognized William II's claim to the English crown, so that was cause enough for William to gather an army and press his claim by right of conquest. The shift in power was decided suddenly at the Battle of Hastings in 1066. Due to his invasion's success, we now style him as William the Conqueror.

Levant - The Normans took up arms when Pope Urban II called the first crusade. Bohemond (a Norman) set up the Principality of Antioch after the crusaders conquered it. Like the conquest of England, it was spurred by Papal consent followed up with the right of conquest and the shift in power was rather quick after the previous rulers were defeated.

Southern Italy - Unlike the first two, this was a gradual and decentralized conquest. Due to the instability and mix of powers in Southern Italy, Norman mercenaries ventured there to earn a small fortune. The issue being Rome was in a precarious position being very close to the Eastern Roman controlled portions of Southern Italy, a few fractured Italian dukes, and Islamic controlled Sicily. This did not make the Pope comfortable. As a result, more and more Normans began military operations against all three aforementioned powers. Over time they carved out lands for themselves (a few counties held by independent Norman families) and it took time for one clear Norman power to develop, the Hauteville dynasty which included Robert Guiscard. By then, the Pope had actually become fearful of the powerful Norman holdings in Southern Italy and they would eventually clash and struggle politically for decades. The Normans had managed to conquer and absorb all of Southern Italy from the Arabs, Eastern Romans, and Italians.

Richard Huscroft The Norman Conquest

R. Allen Brown The Normans

For further reading online you can wiki the two Norman conquests and also the First Crusade.