The largest campaign to reclaim the west was undertaken under the reign of Justinian in the 6th century, most famously under the leadership of General Belisarius. The campaigns were successful in reclaiming Italy, as well as the southern coast of Iberia and some former Vandal-held North African possessions.

A mixture of warfare and plague meant that these possessions were impoverished and could provide little back to Constantinople, so ultimately the wars probably weakened Byzantium's position with respect to more proximal threats (i.e., Persia -- and, in the longer term, the Arabs). Nonetheless, they held onto the Exarchate of Ravenna until the 8th century, and held onto some Southern Italian possesssions until the 11th century.

In addition to the Justinian campaigns, the Byzantines periodically turned their attention to southern Italy (the Catepanate) and Sicily during the 11th C. Their grip on much of the Catepanate had been tenuous since the arrival of the Lombards in the 8th C but Bari and Calabria remained Greek strongholds. The area saw intermittent warfare as allegiances shifted and the Byzantine governor nibbled away at the Lombard principalities.

Sicily was a more extreme case, slipping entirely out of Byzantine hands during the 10th C. This was itself a protracted affair that took decades but it wasn't until 1038 that the Constantinople was able to launch a full-scale invasion to reclaim the island, under the command of the memorable George Maniakes. That ultimately failed and is, in retrospective, more notable for helping to launch the Italian careers of the de Hauteville brothers. The latter would later have to face some Greek armies of their own in Italy.

So Italy remained a priority for the Empire throughout the centuries but the constant threats faced on other fronts (most obviously the Bulgars and Turks) consistently hindered its ability to defend the provinces. There were semi-regular campaigns up to and beyond the end of Byzantine Italy but, for various reasons, none were successful.