Note that while singular they is semantically singular, it is syntactically plural; thus the singular they takes plural verb forms. While this may seem odd, it is no different from the use of you, which always takes a plural verb form, whether one is addressing one or several persons. However, a singular reflexive "themself" has been heard from time to time. By contrast, singular they still takes singular nouns, as in 'If someone is flying a plane, then they are a pilot'.
Although many attack this usage as an aberration introduced for reasons of political correctness, singular they has a centuries-long history of usage. Several famous authors have used it in their writing, including Jane Austen and William Shakespeare. However, it is now becoming more popular, and this may be traced to the rise of feminism.
Many grammar and usage guides state that singular they can only be used to refer to an indeterminate person, but it cannot generally be used to refer to a person identified as a particular unique individual, even if that person's gender is unknown. For example, one might say "A person might find themself in a fix" but not "The doctor might find themself..." In the latter case, the most usual thing to do is to recast the sentence in the plural ("Doctors might find themselves...") or second person ("If you're a doctor, you might find yourself..."). On the other hand, singular they is occasionally used to refer to an indeterminate person whose gender is known: "No mother should be forced to testify against their child."
Not all people agree with this view. Many conservative grammarians view singular they as inherently ungrammatical. Others feel that there is no reason not to extend singular they to include the transgendered and specific people of unknown gender. This debate is tied in with wider issues of political correctness and equal rights, as well as the extent to which language influences thought (see the Sapir-Whorf Hypothesis).