The level of safety in Saudi Arabia in general is high. That said, petty theft does occur in the cities so, as in any other city in the world, you should be on your guard and keep valuables out of sight. Women must take the same safety precautions as they would anywhere else in the world.
Further, security is very tight in the expatriate compounds and visitors are subject to strict checks before being allowed entry. Accordingly, crimes such as burglaries tend not be a concern.
If you wish to explore the countryside of Saudi Arabia it is a good idea to go with a reputable guide who speaks Arabic, as there are small pockets of the country that it is not advisable to visit unaccompanied.