On one occasion I have noticed an interesting usage of their pronoun on a website: X has changed their profile photo. “Has” and “their”? The important fact may be that it was a Russian website translated to English so the original possessive pronoun was probably своё. This Russian form provides us no information about the gender of possessor (it tells us about the gender of the possessed object); probably this is the reason of using a singular gender-neutral pronoun in English translation.
The singular “they” is not a product of modern-era political correctness, but an old phenomenon that occurred in English since the 14th century. However, until recently it was not accepted by traditional prescriptivist grammarians and its correctness is still disputed; on the other hand, the English Wikipedia provides us in the article about Singular They (including its derivatives: their etc.) an example from such a formal text like... the Canadian Criminal Code:
if a peace officer has reasonable grounds to believe that, because of their physical condition, a person may be incapable of providing a breath sample...
(subparagraph 254(3)(a)(ii)); an amendment form 2008
The article quotes also, for instance, George Bush:
If anyone tells you that America's best days are behind her, then they're looking the wrong way. President George Bush, 1991 State of the Union Address
The rival pronoun used in situation when we do not know the gender of person we are talking about, is generic he:
No one shall be arbitrarily deprived of his nationality nor denied the right to change his nationality. — Article 15, Universal Declaration of Human Rights
However its usage is dropping and some people reject it (because of its alleged sexism), as well as some reject singular they. One another proposition for gender-neutral pronoun, that applies mainly to written English, namely he or she, is also often rejected as clumsy, awkward and overly political correct.