Should I use his, his or her, or their?

Indefinite pronouns are often used to make general observations or broadly inclusive statements. In the past, it was often standard practice to use a singular male pronoun to follow indefinite pronouns. This "generic he" was supposed to refer to males and females generally.
Everyone wants to do his best.

However, concerns over sexist language have created a desire for more gender-inclusive constructions.

Everyone wants to do his or her best.
or
Everyone wants to do their best.

Although the second sentence violates the general rule requiring agreement with the antecedents, many writers and speakers prefer to use forms of they because these forms are not gender specific. This is a common practice, but it is still criticized by grammatical purists.

To avoid the problem, rewrite the sentence so that a "generic he" would not be necessary. Usually, the simplest solution is to replace the singular indefinite pronoun with a plural alternative.

All people want to do their best.