Which general led the byzantine troops against the vandals and ostrogoths, under the reign of emperor justinian?

It was General Belisarius.  He was instrumental in helping Justinian in the campaign against the Sassanids where he won a victory against them in the Battle of Darra and the Battle of Callinicum during the Iberian War.  He then led a campaign against the Vandals where he won the Battle of Ad Decimum and the Battle of Tricamarum.  As a result the enemy surrendered and lost Roman territories were given back to the Byzantine Empire.  He received many honors for his success on this campaign. He also led a campaign against the Ostrogoths but had to be recalled back to deal with threats from the Persians who conquered Syria.