Are women allowed to work in western companies?

The number of opportunities for women is increasing.

Some companies do recruit women. This is expected to  increase in future.

At present business is a "mans" world. But there are many companies/business which are founded and run by women.

it's good to know !

Huge opportunity here, All major companies have women department and apart from construction/engineering, i can't think of any type of business in Saudi Arabia that don't have women department

who's going to cook your husbands dinner if you are at work ??

calm down everyone - JOKE

Opportunities are there, but then there is also a thing that they prefer to take in Saudi females. That is a win win situation for the companies here. there is a large pool of educated Saudi women looking for jobs, they work a lot better than their male counterpart and finally, the companies are able to fulfill the saudiazation clause as well.

HR is one major area where this is a huge no. of opportunities for women in particular.
Good luck!