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Alternatively, from your profile, you can set your presence to Do Not Disturb to block chat notifications and let your contacts know you are currently unavailable. Calls will still come through, however. Learn more about how to control who you receive calls from.
Yes, you can control alerts for specific conversations in Skype.
There could be a few reasons why you might not be receiving notifications:
If you close the Skype app without signing out first, you might still receive notifications. If you want to stop these notifications, you can either disable them from your Settings or sign out before you close the app.
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No emergency calls with Skype Skype is not a replacement for your telephone and can't be used for emergency calling