Australia
Yes. But only new dwellings. Not homes which have previously been occupied. With house prices in Sydney and Melbourne going through the roof, the Abbot Government is cracking down on foreign nationals who breach the rules. Much stiffer penalties of up to $140,000 or three years' imprisonment are on the way. Furthermore, would-be homeowners of foreign extraction will have to pay a minimum $5500 application fee before getting approval for their purchase from Australia's overseas investment authority.
China
Land? No. Properties? Yes. The state issues land-use rights to housing and apartment developers. Foreigners can buy such properties, but must have lived or studied in China for at least 12 months.