to get engaged / to start dating
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Fidanzarsi can mean both 'to start dating' and 'to get engaged' — the distinction depends on context. In modern Italian youth culture, it often simply means entering an exclusive relationship.
Si sono fidanzati durante una cena romantica a Firenze.
They got engaged during a romantic dinner in Florence.
Da quanto tempo siete fidanzati?
How long have you been together / engaged?