The English language makes it relatively easy to write in a gender-neutral way because nouns in general do not have a gender; a gender is only specified when referring to people or animals. Hence, there is only one indefinite article (the) and two definite articles (a or an, see here). Compare this to German, where almost every noun has a clear male or female form that requires to chose the correct definite (der, die, das) or indefinite (ein, eine) article. This has inspired concepts such as the Binnen-I to avoid discrimination. However, there are occasions where even in English you would be tempted to write “he/she” or “his/her” in order to not imply that a person is male or female. As an example, consider the sentence
A reader of this post spent five minutes of his/her time to learn about the singular they.
To avoid such rather ugly constructions, the English language offers the so-called singular they. For our example, it allows us to write
A reader of this post spent five minutes of their time to learn about the singular they.
That’s pretty neat, isn’t it? I suggest you follow this link to see a few more examples. Using the singular they may feel awkward in the beginning but is perfectly correct. (Amusingly, I once overheard a German explaining to a native English speaker that it was wrong.)