Off the west coast of Greece, Octavian's navy under Agrippa defeats Antony and Cleopatra, who escape to Egypt.
But at Actium his fleet held out for a long time against Caesar, and only after it had been most severely damaged by the high sea which rose against it did it reluctantly, and at the tenth hour, give up the struggle.
The victory let Octavian consolidate his power over Rome and its lands. Historians generally see that consolidation and his new titles as the end of the Republic and the beginning of the Empire.
Based on Wikipedia: Battle of Actium
Octavian's victory enabled him to consolidate his power over Rome and its dominions.
historians generally view his consolidation of power and the adoption of these honorifics as the end of the Roman Republic and the beginning of the Roman Empire