1148 -- The Norman Conquest of Africa

The historical boundaries of Normandy. The Cherbourg Peninsula was the first territory conquered by the Vikings in their 9th-century invasion. They developed Cherbourg as a port. After the Anglo-Scandinavian settlement, a new name appeared there in a still Latinised form: Carusburg Castellum (1026-1027, Fauroux 58) then Carisburg (1056–1066, Fauroux 214), Chiersburg (William of Jumièges, v. 1070), Chieresburg (Wace, Roman de Rou, v. 1175). Carusburg would mean "fortress of the marsh" in Old Norse kjarr (marsh), and borg (castle, fortified town) or "city of the marais" in Old English ker (bog) and burgh (town). 

820 circa -- First Viking raids along the Seine in France.

911 -- Founding of the County of Normandy. French king grants the raiding Viking Northmen (Normans) a fiefdom on the lower Seine along the English Channel.

The exact date of the treaty establishing Norman control is unknown, but it was likely in the autumn of 911. By the agreement, Charles III, king of the West Franks, granted to the Viking leader Rollo some lands along the lower Seine that were apparently already under Danish control. Whether Rollo himself was a Dane or a Norwegian is not known. For his part, Rollo agreed to defend the territory from other Vikings and that he and his men would convert to Christianity.

999 -- Earliest record of  Normans in southern Italy (inserting themselves there into the conflict between the Byzantines and the Lombards)

Unlike the Norman conquest of England (1066), which took a few years after one decisive battle, the conquest of southern Italy was the product of decades and a number of battles, few decisive. Many territories were conquered independently, and only later were unified into a single Norman state (Kingdom of Sicily). Compared to the conquest of England, it was unplanned and disorganised, but equally complete [WP].

...possibly following pilgrimages to the Holy Land. Some Normans are hired as mercenaries in the wars between the Germanic Lombards and the Eastern Roman Empire for control of the Italian peninsula. Many origin stories of their arrival abound, including this one from Amatus of Montecassino: 

Norman pilgrims returning from the Holy Sepulchre in Jerusalem via Apulia stayed with Prince Guaimar III in Salerno. The city and its environs were attacked by Saracens from Africa demanding payment of an overdue annual tribute. While Guaimar began to collect the tribute, the Normans ridiculed him and his Lombard subjects for cowardice, and they assaulted their besiegers. The Saracens fled, booty was confiscated and a grateful Guaimar asked the Normans to stay. They refused, but promised to bring his rich gifts to their compatriots in Normandy and tell them about possibly lucrative military service in Salerno. Some sources have Guaimar sending emissaries to Normandy to bring back knights, and this account of the arrival of the Normans is sometimes known as the "Salerno (or Salernitan) tradition". 

1009-1022  -- Lombard revolt against Eastern Roman rule in southern Italy, beginning in the city of Bari.

1016 -- Pope Benedict sends Normans to aid the Lombards. 

1019 -- Some Normans begin fighting for pay of the Eastern Roman Empire, against the Lombards in Italy. Norman-garrisoned fort is established in the mountains under Byzantine authority.

1020 -- Pope Benedict, alarmed by Normans now fighting for the Eastern Roman side, goes north to Germany, seeking the help of Emperor Henry II (last of the Ottonian line).

1022 -- Intervention of Western Roman imperial army under Henry II, in support of the Lombards, ends the Lombard-Byzantine war, but fails to retake the Byzantine-allied Norman citadel.

1024 -- Normans begin engaging in mercenary activities in southern Italy for and against the Duke of Naples. Naples is a Byzantine tribute state.

1037 -- Western Emperor Conrad II bestows the Count of Avesso on the Norman leader Ranulf.

1042 -- The Norman William of Hauteville obtains title Duke of Apulia.

1054 -- The Great Schism. Separation of the Eastern (Constantinople) and Western (Rome) Churches. (Question: is it possible to examine the Great Schism without the context of Norman intervention into the Lombard-Byzantine conflict in Italy, and the fact that they profited greatly by keeping this conflict alive?)

1061 -- Normans begin conquest of Arab-controlled Sicily.

1071 -- Roger I (Roger Bosso) (Hauteville) becomes the first Count of Sicily, under authority of his brother, the Duke of Apulia

He was a member of the Norman House of Hauteville, and his descendants in the male line continued to rule Sicily down to 1194.

1071-1086 -- Composition of the Salerno tradition, a record of the Normans in southern Italy. It was first recorded by Amatus of Montecassino in his Ystoire de li Normant

1130 Dec 25 -- Roger II, son Roger I, is crowned the first King of Sicily, uniting Norman controlled lands in southern Italy and the island of Sicily.

1148 -- Roger II invades Al-Ifriqiya (Kingdom of Africa), establishing a Norman kingdom. The Berber Zirid Dynasty is overthrown. Roger takes the title King of Africa, a realm which includes parts of Tunisia and western Libya. Algiers remains outside Norman control, and under Berber authority.

The Mediterranean realms of the Normans in the 12th century AD