The French foreign legion is the most well-known. There are other countries that "hire" non-citizen soldiers as well.
You might be surprised to know that the U.S. military has a large number of foreign nationals in its ranks: Mexicans, Guatemalens, Nicaraguans, Chileans, Venezuelans, Phillipinos, Hungarians, Czech Republic, Russians, Koreans, etc.
The difference being is that the U.S. military does not put them all in the same unit (like the French foreign legion does) but rather spreads them out across all infantry units.
Other countries with sizeable numbers of foreign soldiers: Russia, South Africa, Egypt, Lebanon, China, Singapore, UAE, Saudi Arabia, Pakistan, Iraq.
The current government of the Republic of Iraq employs almost 50,000 foreign troops (Germans, Russians, French mostly) to provide security, force protection, training, etc. This is of course apart from the 140,000 U.S. soldiers there now.