The Colosseum was built:
1) To provide a replacement for the first part-stone amphitheatre (arena) for gladiatorial games in Rome which had been inaugurated by Augustus in 29-30 BC and which burned down in 64 AD.
2) to reclaim the area in front of Nero's enormous palace (the Domus Aurea, Golden House) for public use.
3) to celebrate the Roman victory against the Jews in the revolt of the First Roman-Jewish War (66-73 AD).
4) To show the grandeur of Emperor Vespasian, whose rule was initially shaky.