Grazie!

Italian Expression	English Translation (Literal / Common Meaning)
Grazie infinite.	Infinite thanks.
Grazie di cuore.	Thanks from the heart.
Molte grazie.	Many thanks.
Grazie mille.	Thanks a thousand / Thanks a lot.
Mille grazie.	A thousand thanks.
Grazie tante.	Thanks a lot. (Can be sincere or sarcastic depending on tone.)
Tante grazie.	Many thanks.
Grazie di tutto.	Thanks for everything.
Ti devo un favore.	I owe you a favor.
Tu non sai quanto ti sono grato/a.	You don't know how grateful I am to you.
L’ho apprezzato [veramente].	I [really] appreciated it.
Ti ringrazio, sei molto gentile.	I thank you, you are very kind.
Ti sono debitore/a.	I am indebted to you.
Ti sono obbligato/a.	I am obliged to you.
[Molto] obbligato/a.	[Much] obliged.

Italian Expression	English Translation (Literal / Common Meaning)
PREGO	You're welcome. / Don't mention it. (The most standard and formal response.)
FIGURATI	Don't mention it. / It was nothing. (Informal, singular "tu" form)
SI FIGURI	Don't mention it. / It was nothing. (Formal, singular "Lei" form)
DI NULLA	It's nothing. / Not at all.
DI NIENTE	It's nothing. (Slightly more casual than "di nulla")
NON C'È DI CHE	It's nothing. / Don't mention it. (Literally: "There isn't anything [to thank for].")
E DI CHE	What for? (Literally: "And for what?" – informal)
È STATO UN PIACERE	It was a pleasure.
NON C'È PROBLEMA	No problem.
È IL MINIMO	It was the least I could do. (Literally: "It's the minimum.")
CI MANCHEREBBE	Of course. / I wouldn't dream of not helping. (Literally: "That would be lacking" – implies it's unthinkable not to do it.)