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By default, the ActionMessages and ActionErrors are saved into the Request object and therefore go away after the response is rendered on the client. However, you can manually place these error messages into the session and then when neccessary, get them out and display them to the user. It'll take a little more work on your part because you'll have to remember to manually remove them after you display them to the user. However, this is pretty straightforward and is what most developers do to solve this problem.

Chuck