The MySQL server mysqld can be started and run by any user. In order to change mysqld to run as a Unix user user_name, you must do the following:
Stop the server if it's running (use mysqladmin shutdown).
Change the database directories and files so that user_name has privileges to read and write files in them (you may need to do this as the Unix root user):
shell> chown -R user_name /path/to/mysql/datadir
If directories or files within the MySQL data directory are symlinks, you'll also need to follow those links and change the directories and files they point to. chown -R may not follow symlinks for you.
Start the server as user user_name, or, if you are using MySQL Version 3.22 or later, start mysqld as the Unix root user and use the --user=user_name option. mysqld will switch to run as the Unix user user_name before accepting any connections.
To start the server as the given username automatically at system startup time, add a user line that specifies the username to the [mysqld] group of the /etc/my.cnf option file or the my.cnf option file in the server's data directory. For example:
[mysqld] user=user_name
At this point, your mysqld process should be running fine and dandy as the Unix user user_name. One thing hasn't changed, though: the contents of the permissions tables. By default (right after running the permissions table install script mysql_install_db), the MySQL user root is the only user with permission to access the mysql database or to create or drop databases. Unless you have changed those permissions, they still hold. This shouldn't stop you from accessing MySQL as the MySQL root user when you're logged in as a Unix user other than root; just specify the -u root option to the client program.
Note that accessing MySQL as root, by supplying -u root on the command-line, has nothing to do with MySQL running as the Unix root user, or, indeed, as another Unix user. The access permissions and usernames of MySQL are completely separate from Unix usernames. The only connection with Unix usernames is that if you don't provide a -u option when you invoke a client program, the client will try to connect using your Unix login name as your MySQL user name.
If your Unix box itself isn't secured, you should probably at least put a password on the MySQL root users in the access tables. Otherwise, any user with an account on that machine can run mysql -u root db_name and do whatever he likes.
On Windows, you can install MySQL 4.0.17 and MySQL 4.1.2 as services as a normal user. (Older MySQL versions required you to have administrator rights; This was a bug introduced in MySQL 3.23.54).