Her brothers are both teachers. (after an auxiliary)

Their children both play the piano. (before a verb)

Both her brothers are teachers. (before a noun if you want to insist)

Both their children play the piano.

She has seen both films.

She has seen both the films. (with the definite article but mind the structure)

Both of those books are interesting.(with 'of')

or

Both those books are interesting. 

I saw both of them. (with 'of' before a pronoun)

I met both of them.

He has read both books.

He has read the two books. ('both' may be replaced by 'the two')

The two countries are different. (here you insist on the difference)

The two houses are different.

He is both handsome and clever.

She likes both English and French.

He is good at both mathematics and music.

Beginners
English exercise "Both" created by lucile83 (03-03-2008) with The test builder
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1. She is both ofboth intelligent and pretty.
2. The houses are boththe two too expensive.
3. My children both ofboth went to London last year.
4. Where are the children?..They are here, I can see bothboth of them.
5. She will pass her exam, she is good at the twoboth French and English.
6. I know their twins, the twoboth girls are very different.
7. Did you see Jane and Mike yesterday? ... yes I met bothboth of them at the post-office.
8. Their dogs are awful ! they the twoboth made the kitchen dirty.
9. I like none of these actors, do you?... oh yes, I like both ofthe two them.