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Stare vs. rimanere vs. restare

The Italian verbs stare, rimanere, and restare can all be translated as to stay or to remain, but they have distinct nuances. Stare often relates to a physical location or a temporary state of being, while rimanere and restare are very similar and emphasize the act of not leaving or what is left over.

Stare

A1
Stare primarily means to stay in a physical place or to be in a certain state or condition. It is also famously used to form the progressive tense (e.g., I am doing something) and to express health or mood, which are uses not shared with the other two verbs.
Stasera sto a casa a guardare un film.
(Tonight I'm staying at home to watch a movie.)
Come stai? Sto molto bene, grazie.
(How are you? I am very well, thank you.)
Per favore, stai fermo un momento!
(Please, stay still for a moment!)
Non mi disturbare, sto lavorando.
(Don't bother me, I am working.)
I miei zii stanno a Roma.
(My aunt and uncle live/stay in Rome.)

Rimanere

A2
Rimanere means to remain or to be left over. When it means to stay, it often emphasizes the idea of staying behind while others leave, or the result of a situation. It is very frequently used for things that are left.
Tutti sono andati via, ma io sono rimasto.
(Everyone left, but I remained.)
Dopo la festa, sono rimasti solo due pasticcini.
(After the party, only two pastries were left.)
Sono rimasto senza parole quando ho sentito la notizia.
(I was left speechless when I heard the news.)
Rimango a Milano per un'altra settimana.
(I'm remaining in Milan for another week.)
Mi è rimasta in mente quella canzone.
(That song has remained in my mind.)

Restare

A2
Restare is very similar to rimanere and often interchangeable when meaning to stay in a place. It can sometimes imply a more active decision to stay put and is also used to mean to be left.
Resto qui con te, non ti preoccupare.
(I'll stay here with you, don't worry.)
Non resta molto tempo, dobbiamo sbrigarci.
(There isn't much time left, we have to hurry.)
È restato deluso dal risultato.
(He was left disappointed by the result.)
Se piove, restiamo in casa.
(If it rains, we'll stay at home.)
Cosa resta da fare adesso?
(What remains to be done now?)

Summary

In short: Use stare for temporary location (sto a casa), for conditions (sto bene), and for the progressive tense (sto mangiando). Use rimanere and restare when you mean to remain or to be left over; they are largely interchangeable in this context (rimango/resto qui). Rimanere often highlights what is left after a change, while restare can sometimes imply a more conscious decision to stay.