We have no historical record outside of the Old Testament for a man called Moses leading a large group of Israelites out of Egypt.
This is particularly odd considering the fact that the Egyptians would generally have recorded such a mass exodus of slaves which would have impeded their building efforts. Some Christian apologetics argue that the Egyptians would not have recorded such an event because they would have been embarrassed by this diplomatic defeat. This is highly unlikely considering the fact that there would have been 2.5 - 3 million Israelites leaving Egypt at once based on an extrapolation of Numbers 1:46. During this period there were only 3.5 - 4 million Egyptians in Egypt. Surely such an exodus would have left a huge gap in their economy? They would have had to account for this in some way even if they muddled the facts to make it appear more favourable to their Pharaoh.
In addition as Moses was in essence an Egyptian prince, he would have been mentioned in their royal records. So where did this story come from? According to Hebrew scholars the first five books of the Bible were produced by a scribe (possibly Ezra) around the 5th century BCE. The were redacted from earlier sources which probably dated back to the 7th century BCE. See Richard Friedman's "Who Wrote the Bible" for more information on this. Thus the story of Moses was first written as it is today following Israel's enslavement in Babylon which lasted three generations. This is important as we have to account for a certain amount of Babylonian influence on the Israeli culture during this time.
There are two possible sources for the story of Moses and the most probable explanation is a combination of the two:
King Sargon and Moses
Most of us are familiar with the story of Moses' birth. Moses was born during a time when all the first born males were being killed by Pharaoh's men to prevent a Jewish revolt. Moses' mother makes a basket made out of bulrushes as sealed it with bitumen and pitch. She then places the baby in the basket and hides him in the river amongst the reeds. Later the Pharaoh's daughter finds Moses and adopts him.
Archaeologists have found cuneiform tablets dating back to 1000 BCE which describes the story of neo-Babylonian King Sargon who lived around the third millennium BCE. These tablets also tell the story of how this mighty ruler was saved as a baby when his mother made a basket out of "rushes and sealed it with tar". He was also placed in a river and found by a princess who later raised him. Thus we have situation where the Israelites enslaved in Babylon adopted this story and possibly merged it to an even earlier legend that we will address later.
It's worth noting that the cuneiform tablets from Babylon also contained the Epic of Gilgamesh which was an earlier version of Noah's flood and the Code of Hammurabi which was a prototype of Moses' law. It even included the famous "eye for an eye" quote. The fact that these Babylonian folk tales are much older than their Hebrew counterparts and the fact that we find no evidence for these stories in Hebrew literature before the Babylonian enslavement, are indications that the Hebrews "borrowed" them from the Babylonians. You can find more information on these tablets here, here and here.
Ah-mose and the Hyksos
Now that we have established that the birth and law of Moses was adopted from earlier Babylonian sources we have to look at how the actual exodus out of Egypt. The actual exodus is discussed here, however we do need to look at Moses' role in this story.
As we have already stated, there is no evidence for Israelite enslavement in Egypt or any mass exodus out of Egypt. So where did this legend come from?
When leading archaeologists discovered evidence for a large group of people fleeing Egypt through the Sinai desert, they were surprised to find that these were not Israelis crossing the desert, but a group of people called the Hyksos. They were a group of Asiatic and Semitic people who occupied Egypt during the 17th and 16th centuries BCE. An Egyptian leader by the name of Ah-mose (also known as Amosis) was the founder of the Eighteenth Dynasty in 1550 BCE.
When he came to power he managed to expel the Hyksos from Egypt and thus liberated the entire Egypt. His men then chased the Hyksos through the Sinai desert and through Canaan. It's worth noting that archaeologists have found evidence for this large migration by the Hyksos as well as the wanderings of small nomadic groups through the Sinai desert, but no evidence has been found that would support an exodus by the Israelites consisting of 2.5 - 3 million people over a period of 40 years.
Finally Ahmose reasserted Egyptian rule over areas such as Canaan and Nubia to ensure that such an occupation by the Hyksos would never occur again. This enforcement included the setting up of Egyptian military outposts which could be reached within one day in any direction in the Sinai desert. This is interesting considering that the Israelites were supposed to have wandered this small desert for 40 years, however they never stumbled on to these Egyptian military outposts.
This story of Ahmose became legendary and by the time that the Jewish history was written a thousand years later, the story of Ahmose expelling the Hyksos out Egypt and establishing his rule in Canaan was changed to Moses leading the Israelites out of Egypt and conquering Canaan. This could have resulted from some of the the fleeing Hyksos settling in Canaan. Some Christian Apologetics use the story of the Hyksos as evidence of the Exodus and claim that the Hyksos were in fact the Israelites, however we have shown how the Hyksos could not have been the Israelites (see question 'Who were the Hyksos people?'.
So in conclusion our historical and archaeological records show no trace of a Moses character ever existing or leading the nation of Israel out of Egypt. Instead, all of the evidence outside of the Old Testament points to the expulsion of the Hyksos by Ahmose as well as the additional neo-Babylonian folk stories developing into a composite mythical character. This character helped give Israel a new national identity after their enslavement in Babylon.