(My answer to this question at Quora.com)
As Ricardo Almeida points out, the story of the Exodus bears much resemblance to the historical explusion, around 1550 BCE, of the “Asiatics” (Canaanites) who had settled and eventually took over and ruled northern Egypt for well over four hundred years.
Here’s Flavius Josephus, quoting an Egyptian historian of the 3rd century BCE (Manetho):
“These people, whom we have called kings before, and shepherds too, and their descendants,” as he  says, “held Egypt for five hundred and eleven years. Then,” he says, “the kings of Thebes and the other parts of Egypt rose against the shepherds, and a long and terrible war was fought between them.”
He says further,
By a king, named Alisphragmuthosis , the shepherds were subdued, and were driven out of the most parts of Egypt and shut up in a place named Avaris, measuring ten thousand acres.” Manetho says, “The shepherds had built a wall surrounding this city, which was large and strong, in order to keep all their possessions and plunder in a place of strength.
Tethmosis , son of Alisphragmuthosis, attempted to take the city by force and by siege with four hundred and eighty thousand men surrounding it. But he despaired of taking the place by siege, and concluded a treaty with them, that they should leave Egypt, and go, without any harm coming to them, wherever they wished. After the conclusion of the treaty they left with their families and chattels, not fewer than two hundred and forty thousand people, and crossed the desert into Syria. Fearing the Assyrians, who dominated over Asia at that time, they built a city in the country which we now call Judea. It was large enough to contain this great number of men and was called Jerusalem.
Over the following centuries, some of those Canaanites, who traced their ancestry back to one Jacob a.k.a. Israel, commemorated the Exodus from Egypt by making it into a central part of the Israelite mythology and national identity. Naturally, certain changes crept in: such as the idea that they had been oppressed slaves all their time in Egypt, instead of settlers who ruled over northern Egypt for over five hundred years (the only vestiges of which is the story of Joseph being elevated to viceroy, and Moses being raised as an Egyptian prince), and that all those who had been expelled were descendants of Jacob/Israel, etc.
The most compelling piece of evidence, though, is a seemingly innocent piece of graffiti made at the time of the exodus (1550 BCE) on a statue of the Egyptian goddess Hat-hor at the jade mines in western Sinai (a place today called Serabit el-Khaddem), which, in old but clearly decipherable Canaanite writing, says mt[t] lba’alat, meaning A gift for Baalat in Canaanite/Hebrew:
Baalat was the generic Canaanite term for “goddess”; Hat-hor was the patron Egyptian goddess of mining, but originally a Canaanite goddess. The fact that these Canaanites were desecrating one of her statues (the graffiti itself was a desecration, but the “gift” was something unmentionable that they deposited on top of it), means that they themselves were no longer believers in that goddess—i.e., they were believers in the nascent monotheistic religion that we now call Judaism.
This story—which is mind-blowing on many levels, because it also heralds the dawn of Western literacy, with all that that entails—will be told in full in my forthcoming story of Sherlock Holmes and the Case of the Mistress of History.