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How will I know when someone sends me an instant message in Skype for Mac OS X?
If you are signed in to Skype and you receive an instant message when the conversation window is not open, a numbered badge will appear next to the contact in the sidebar. The number displayed indicates the number of unread messages in the conversation.
A red numbered badge will also appear on the Skype icon in the dock. This indicates the number of conversations with unread messages.
If you receive a new instant message from participants in a voice or video call, you can see this icon:
Click it to read the instant message.
Learn more about sending instant messages.