Singular possessive

The singular possessive is normally formed by adding ’s

the author’s work
the patient’s reaction
Smith’s paper
Newton’s laws

Names that end in ‑s, ‑z or ‑x

For names that end in ‑s, ‑z or ‑x, you can use an apostrophe either with ‑s or without ‑s.

In many cases both are possible:

Bayes’ theorem or Bayes’s theorem
Pepys’ diary or Pepys’s diary
Yeats’ poetry or Yeats’s poetry

However, use an apostrophe with ‑s if the possessive sounds unclear without ‑s:

Gauss’s law
Tallis’s music
Confucius’s analects
Linnaeus’s taxonomy
Erasmus’s work
Dylan Thomas’s Under Milk Wood
Lenz’s law

And use the apostrophe without ‑s if the possessive sounds or looks awkward with ‑s:

Archimedes’ principle
Achilles’ heel
Cervantes’ Don Quixote

Alternatively, you can paraphrase:

the reign of King James
the works of Tacitus
the Linnaean system
Marxist theory

Plural possessive

Most plural nouns simply add an apostrophe after the ‑s to form the possessive:

the authors’ work
patients’ reactions
participants’ feedback

Nouns whose plural does not end in -s

Plural nouns like men, women and children add ’s

men’s room
women’s rights
children’s hospital

Compound possessive

In compound possessives, add ’s to the last word of the compound:

the Security Council’s decision
William the Conqueror’s victory
Watson and Crick’s breakthrough

Possessive pronouns

The possessive pronouns do not have an apostrophe:

his ours whose
hers yours
its theirs

But one’s does have an apostrophe:

One’s immediate reaction is …

Note: it’s with an apostrophe is not a possessive pronoun, it is short for it is. Similarly, who’s with an apostrophe is short for who is. Neither is used in formal writing.