The first stone amphitheatre was built in Pompeii in 70 BCE (Pompeii was a colony for Roman military veterans); it took the Romans a long time to build a stone purpose built amphitheatre in Rome as they were worried about moral corruption if they built stone theatres and amphitheatres in Rome. Games were held in the Forum Boarium and in the Forum Romanum, with temporary stands being set up and tunnels being built under the Forum Romanum by Julius Caesar to funnel gladiators and animals into these temporary arenas. The most notorious temporary arena was that built by Gaius Scribonius Curio, one of Caesar’s supporters, in 52 BCE. Struggling to make his mark after Marcus Aemilius Scaurus had built a fabulously expensive theatre during his aedileship in 58 BCE (it had an entire level built out of glass), he went for a theatre that turned into an amphiteatre:
Curio (who died during the Civil War while fighting for Caesar) had no hope of outdoing Scaurus in expensive decorations in his games for his father…so he had to think hard and come up with some new scheme. It’s a valuable lesson for us to know what he came up with and to be pleased with our values and, in a shift from what is usual, to call ourselves [moral] ancestors. He constructed two large wooden theatres right beside each other, each of which pivoted on a revolving point. In the morning each one hosted a play, and each half faced away from the other so that the plays did not drown each other out. And, then, suddenly each one revolved (and the sources say that after the first few days some spectators kept sitting as it did so) and the corners met and the whole became an amphitheatre in which he gave gladiatorial battles – although the gladiators were less for sale than the Roman people as they whirled around.
Pliny the Elder, Encyclopaedia 36.117. (Pliny then goes on a long rant about the sheer immorality of this.)
Eventually Statilius Taurus built a stone amphitheatre (probably in the Campus Martius) in 29 BCE, at the urging of Octavian (later Augustus); it burned down in the Great Fire of Rome in 64 CE. It was also never all that satisfactory; Caligula started building a replacement, but it was never finished. Nero built one of wood. And it was the Flavians, the dynasty that replaced the Julio-Claudians after the suicide of Nero, who built the Colosseum on the private lake attached to Nero’s Golden House.
They financed its building with the funds from their victory in the First Roman-Jewish War (66-73). (For more on amphitheatres in Rome, go to this page from Lacus Curtius, which also supplies references.) And that was the first time that Rome had an amphitheatre to match her power. There was another significant amphitheatre in Rome: the AmphitheatrumCastrense. However, it was built long after the Colosseum.
Amphitheatres outside Rome
There were amphitheatres all around the Roman world, east and west. Many of the most spectacular remains are now in North Africa, the one below from El Jem in Tunisia:
Those who could not afford to build amphiteatres, often used theatres for gladiators and beast hunts: the Theatre of Dionysus in Athens was used for this purpose.The philosopher and holy man Apollonius of Tyana (1st century CE) attacked the Athenians for their fondness for gladiatorial munera and their use of the Theatre of Dionysus for such shows:
He also corrected the following abuse at Athens. The Athenians ran in crowds to the theatre beneath the Acropolis to witness humans killed, and the passion for such gladiatorial sports was stronger there than it is in Corinth today, for they would buy adulterers, fornicators, burglars, robbers, and kidnappers and similar rabble for large sums, and then they took them and armed them and made them fight with one another. Apollonius then attacked these practices, and when the Athenians invited him to attend their assembly, he refused to enter a place so impure and reeking with gore.
And he this said in a letter to them; he said that he was surprised that “the goddess Athena had not already fled from Acropolis, when you shed such blood before her eyes. For I suspect that soon, when you are conducting the pan-Atheniac procession, you will no longer be content with bulls, but will be sacrificing hecatombs of men to the goddess. And you, Dionysus, do you attend their theatre after such bloodshed? And do the wise among the Athenians pour libations to you there? No! Depart, Dionysus! Holier and purer is your Cithaeron.”
Philostratus, The Life of Apollonius of Tyana 22
Rome was a very status conscious society, with a small elite of senators and equestrians. Initially, seating in the arena was indiscriminate – everyone sat where they could, though presumably the editor, the person giving the show, and his friends and family had seats in their control. Gradually seating in the theatres and at munera became more stratified, with seating restrictions and rules becoming legally binding (although seating in the Circus Maximus was always more chaotic, probably a feature of its larger size (it sat 250,000 people to the Colosseum’s c.50,000) and an inability or unwillingness to enforce seating restrictions on that large a crowd. In 194 BCE senators got special seats at the ludi Romani; in 67 BCE the Lex Roscia created a special seating section (of 14 rows) in the theatres for equestrians. The tribune of the plebs, Lucius Otho Roscius, was proposed this law, was booed by the crowd for this; the equestrians, on the other hand, applauded him wildly. Under the Republic gladiatorial and theatrical shows were held in temporary structures built in the forums; the seats were also temporary and seating had to be flexible to some degree. However some had special seats because they had given land for buildings or for other reasons. For example, in 184 BCE the Maenii were granted special places to watch spectacles on the Basilica Porcia. The Emperor Augustus brought in the most detailed seating restrictions yet. Rather than talking about these, I direct you to the University of York’s Jonathan Edmunson’s article on seating and why it matters.
 An annual festival with athletic games in honour of the goddess Athena.