"Gracefully" killing a process

Right now I am using Process.Kill() to kill a process. Is there a way though, instead of just killing it immediately, that I can like send a message to the process instructing it to close so that it can gracefully clean up and shut down. Basically, I'm looking for the equivlent to just clicking the red X in the upper right hand corner, which I believe DOES send a message to the application requesting a shut down.