Capturing program return codes in a bash script?

How do I determine if an command completed successfully when run from a
bash script to decide what the script should do next?

Working on a mysqldump backup script which works pretty nice... but
doesn't really care whether the database dump dies, errors out or
completes successfully.

I figure that's kind of important for a backup script.

-- 
"Remain calm, we're here to protect you!"