Current law restricts property ownership, business or residential, to Kuwaiti nationals.
Recently, however, Kuwait's Parliament has started to draft legislation allowing foreign ownership of residential and commercial property by non-Kuwaitis. Even then, prospective property owners will have to adhere to strict guidelines, like:
Additionally restrictions include, owning only one property of no more than 350sqm. Ownership does not guarantee immediate residency in Kuwait. Finally, if a property owner leaves Kuwait for permanently, they will be required to sell their property.
Below is a list of prominent real estate companies in Kuwait.
Several papers have a helpful classified section. Occasional property listings do come up. Your best bet, however, is to contact a real estate agent and ask for help.
Kuwait General Administration of Customs is an authorized auction house. From time to time, KGAC has public auctions. A list of items being auctioned off is available on their website in Arabic, as too are rules and guidelines for participants.