Living in Rome, puzzare is a word I need a lot – every time I pass an overflowing rubbish bin, in fact.
It means, quite simply, 'to stink'.
Questo maglione comincia a puzzare, bisogna lavarlo.
That sweater is starting to stink, it needs washing.
Il tuo alito puzza di aglio.
Your breath stinks of garlic.
It's believed to come from the same Latin root that gave us the English word 'putrid', and it's about as unpleasant.
Try saying it aloud: “poo-tsar-reh”. Just pronouncing it forces you to curl up your lip in disgust.
While puzzare is the verb, puzzolente is the adjective ('stinky') and una puzza is the noun ('a stink' or 'stench').
NB: though your dictionary will probably direct you to the masculine form, un puzzo, personal experience and internet consensus suggests that the feminine version is more common, in spoken Italian at least.
C'è puzza di fumo in questa stanza.
There's a stench of smoke in this room.
Che piedi puzzolenti che hai!
What stinky feet you have!
The stench you're referring to doesn't have to be literal: much like we say in English that something 'smells fishy', if an Italian says something stinks, they might mean there's something suspect about it.
Tutta questa storia puzza d’imbroglio.
This whole story stinks of fraud.
Il silenzio del capo puzzava a tutti.
Everyone thought the boss's silence was fishy (literally: the boss's silence stank to everyone).
Think about the Italian verb sapere: it means both 'to know' and 'to smell' or 'to taste'. That's why the phrase mi sa ('it smells to me like…') is like saying 'I'm pretty sure that…'
So when something 'stinks', you've got a strong hunch that something's wrong. You can just smell it.
I smell a rat!
Think about the face you make when you smell said rat (or fish): you'd probably curl your nose up, right?
That's why if you say that someone “ha la puzza sotto il naso” ('has a stink beneath their nose'), you're saying they're a snob: they've got their nose stuck up in the air.
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