Pickpocketing is a problem in Egypt's bigger cities, particularly Cairo. Many locals opt not to carry wallets at all, instead keeping their money in a clip in their pocket, and tourists would be wise to adopt this as well. On the upside, violent crime is rare, and you are highly unlikely to physically mugged or robbed. If, however, you do find yourself the victim of crime, you may get the support of local pedestrians by shouting "Harami" (Criminal) while chasing the person who robbed you.
Overall, scams are the main concern in Egypt. Be aware that many Egyptians who starts a conversation with you in Cairo and Luxor want your money. There is a very insidious tactic used where they will "befriend" you, take you around, show you things, even bring you back to their place for dinner, and then they will demand money for it. Basically, if it seems too good to be true, it probably is. Demand prices for absolutely everything, because if you say "I thought it was free!" after the fact you are in for a vicious argument.