‘Keep me posted’ means “please keep me updated on the progress” or “please update me.”
It is a very commonly used expression not only in formal situations but also in everyday conversation.
Since ‘post’ originally means “to post” or “to inform,” “keep me posted” naturally means “please keep me informed of the news.”
It is a safe expression that can be used with close friends, coworkers, and in business emails alike.

3 Everyday English Examples
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Please let me know later how the new project is going. → Keep me posted on your new project.
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Please be sure to let me know when the test results come out. → Keep me posted about your test results.
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Please let me know if the schedule changes. → Keep me posted if the schedule changes.
Examples of Academic and Professional English Expressions
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Please keep me informed if there is any update on IRB approval status. → Please keep me posted on the IRB approval status.
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Could you let me know once the session time is confirmed? → Keep me posted once the session time is confirmed.
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Please let me know if there are any changes regarding the reviewers’ comments. → Keep me posted on any updates regarding the reviewers’ comments.