Wendi Deng gained popularity for springing to her husband's aid when a protester threw a custard pie at him as he appeared before MPs to answer questions about phone hacking in 2011.
At 44, Ms Deng is 38 years younger than Mr Murdoch, who is 82.
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The PM will sound out the possibility of new free trade agreements with Gulf states for once the UK has left the EU.
In the unexpected announcement on Monday, Mr Key said it was the "right time" to leave politics.
As the Government's Brexit appeal comes to the Supreme Court, here's a look at what you need to know.