Short answers, pulled from the story.
The most well-known date for the establishment of the republic is 509 BC. This specific dating emerges from the Varronian chronology, assembled during the late republic by Marcus Terentius Varro.
Roman noblemen led by Lucius Junius Brutus obtained support from the aristocracy and people to expel the king and his family. They created a republic where two consuls are elected annually to rule the city.
Sources such as Livy, Dionysius of Halicarnassus, and Plutarch form the core accounts but lack reliability due to being written centuries after events. T. P. Wiseman argues that many kings and figures were ahistorical inventions of the fourth and third centuries BC.
Modern scholars are sceptical of much traditional chronology because lists of consuls are an unreliable anachronism of the late republic. Resolution is difficult due to the absolute paucity of reliable sources and archaeological evidence showing destruction around 500 BC.
The putative role of Lucius Junius Brutus was commemorated with a statue on the Capitoline Hill holding an unsheathed sword. His name was appropriated during the French Revolution and referenced in The Federalist Papers to inspire democratic movements.