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I can confirm that Amex has this. If you login on the website, you can also get emails / text for each transaction, set thresholds, etc. I've been using this feature for a while now. They also send you weekly statements on your transactions and how much your account has changed from the previous week, etc.

The Amex app on iPhone sends push notifications for changes.

Chase also supports this via SMS, email, or push notification.

Thanks! I was unaware of this feature!

As others pointed out, AMEX has supported this for a while - I get texted almost instantly with any transactions on my card. I believe their mobile app also supports notifications, but I prefer SMS for this.

I only wish my bank (Wells Fargo) supported SMS alerts for transactions - if there's one thing I don't mind getting frequent notifications about, it would be this.