The Triumvirate was established by law in 43 BC as the Triumviri Rei Publicae Constituendae Consulari Potestate ("Triumviri for the Constitution of the Republic with Consular Power", invariably abbreviated as "III Vir RPC"). It possessed supreme political authority; the only other office which had ever been qualified "for the constitution of the Republic" was the dictatorate of Lucius Cornelius Sulla. The only limit on the powers of the Triumvirate was the five-year term set by law.
A historical oddity of the Triumvirate is that it was an effectual three-man dictatorate which included Antony, who in 44 BC had passed a lex Antonia which had abolished the dictatorate and expunged it from the Republic's constitutions. As had been the case with both Sulla's and Julius Caesar's dictatorates, the members of the Triumvirate saw no contradiction between holding a supraconsular office and the consulate itself simultaneously (Lepidus was consul in 42 BC, Antony in 34 BC, and Octavian in 33 BC).
Octavian, who despite his youth had extorted his way to having been named suffect consul (consul suffectus) for 43 BC, had been warring with Antony and Lepidus in upper Italia when they met at Bologna in November that year and agreed to unite and seize power. In order to refill the treasury, the Triumviri decided to resort to proscription; as all three had been partisans of Caesar, their choices of targets were somewhat peculiar. The most notable victim, Marcus Tullius Cicero, who had opposed Caesar and excoriated Antony in his Philippics, came as no surprise, but the proscription of Caesar's legate Quintus Tullius Cicero (the more famous Cicero's younger brother) seemed to be motivated by pure spite. Two Perhaps the most shocking proscription was that of Caesar's legate Lucius Iulius Caesar, Caesar's first cousin, once removed (and Antony's uncle!) and one of Caesar's closest friends.
Octavian's colleague in the consulate that year, Quintus Pedius, died before the proscriptions got underway. Octavian himself resigned shortly after, allowing the appointment of a second pair of suffect consuls (the original consuls for the year, Caesar's legate Aulus Hirtius and Gaius Vibius Pansa Caetronianus, had died fighting on the Senate's side of the first civil war to follow Caesar's death, that between the Senate and Mark Antony himself). This became a broad pattern of the Triumvirate's two terms; during the ten years of the Triumvirate (43 BC – 33 BC), there were 42 consuls in office, rather than the expected 20.
The Caesarean background of the Triumviri made it no surprise that immediately after the conclusion of the first civil war of the post-Caesar period, they immediately set about prosecuting a second: Caesar's murderers Marcus Iunius Brutus and Gaius Cassius Longinus had usurped control of most of the Eastern provinces, including Macedonia, Asia Minor, and Syria. In 42 BC, Octavian and Antony set out to war, defeating Brutus and Cassius in two battles fought at Philippi (both assassins committed suicide). In October 40 BC, the Triumviri agreed to divide the provinces of the Republic into spheres of influence; Octavian -- who had begun calling himself "Divi filius" ("son of a divinity") after Caesar's deification as Divus Iulius ("the Divine Julius") and now styled himself simply "Imperator Caesar" -- took control of the West, Antony of the East, and Lepidus of Africa.
While Antony cemented his hold in the East and reformed the provincial administration (like Sulla's provincial reforms, Caesar's had been quietly ignored after his death), Octavian tightened his grip on the West and nominally oversaw a campaign against the pirate commander Sextus Pompeius (the campaign was actually commanded by Octavian's lieutenant, Marcus Vipsanius Agrippa), which culminated in victory in 36 BC; Agrippa had been consul in 37 BC and had secured the Triumvirate's renewal for a second five-year term.
Like the First Triumvirate, the Second Triumvirate was ultimately unstable and could not withstand internal jealousies and ambitions. Antony cordially detested Octavian and spent most of his time in the East, while Lepidus favoured Antony but felt himself obscured by both his colleagues, despite having succeeded Caesar as Pontifex Maximus in 43. Consequently, Lepidus cooperated in Octavian's campaign against Pompeius (son of Gnaeus Pompeius Magnus) but foolishly attempted to seize control of Octavian's victorious legions; Octavian unilaterally expelled Lepidus from the Triumvirate, but allowed him to retain his Pontificate.
Despite having married Octavian's sister in 40 (Octavian married Antony's stepdaughter Scribonia in 43), Antony openly lived in Alexandria with Cleopatra VII of Egypt and sired children by her. A master of propaganda, Octavian turned public opinion against his colleague; when the Triumvirate's second term expired in 33 BC, Antony continued to use the title Triumvir (Octavian, opting to distance himself from Antony, did not use it). Octavian illegally gained Antony's will in July 32 BC and read it aloud; it promised large legacies to Antony's bastards by Cleopatra and instructed that his body be shipped to Alexandria for burial. Rome was outraged, and the Senate declared war.
Octavian's forces decisively defeated Antony's and Cleopatra's at Actium in Greece in September 31 BC and chased them to Egypt in 30 BC. Both Antony and Cleopatra committed suicide in Alexandria, and Octavian personally took control of Egypt and Alexandria (Egyptian chronologies consider Octavian as Cleopatra's successor as Pharaoh). With the complete defeat of Antony and the marginalisation of Lepidus, Octavian was left sole master of the Roman world, and proceeded to establish the Principate as the first Roman "emperor".
Triumviri Rei Publicae Constituendae Consulari Potestate (43 – 33):