His and her agree with indefinite pronouns like everyone, everybody, each, nobody, no one. This agreement is easy when the genders are clear:

Each of the women's track team members brought her suitcase.

Everybody on the football team must get his uniform cleaned. (Unless it's a co-ed or women's team.)

Many writers object to his as "the universal pronoun" because it seems to exclude women, preferring their to designate either or both sexes. Their, however, is incorrect. You have several options, none of them, alas, ideal.

You can use the pronoun belonging to your own gender except when the meaning clearly calls for a different one, as with:

Each contestant helped the other with her skirt.

You can stick with his in most written English, and use his/her in addressing groups of people, writing reports, letters to the editor, or documents for general distribution. In this case, indicating his/her once at the beginning signals that you mean both sexes and prevents the clumsiness of continual repetition.

You can use her equally with his when discussing hypothetical cases, as this handbook uses both her and his to indicate "the student" or "the writer."

All the picnickers brought their baskets.

(Instead of "Each picnicker brought her basket.")

You can invent your own grammatical term and attempt to get everybody to use it. One such term is ter, another is peep. Both are bizarre.

   Sentence Fragments
   Comma Splices
   Run-on Sentences
   Dangling or Misplaced Modifiers
   Subject-Verb Agreement
   Split Infinitives
   Which, That, It, and Of
   Semi-colons and Colons
   Apostrophes
   His and Her