Ok apparently I do have a problem. =(
This is what I am doing.
Log in to my Debian machine through SSH remotely using a regular user account (not root). I type in “mysql –u root” and it logs me into mysql as if I were root. It doesn’t ask me for a password AT ALL, and I have root privileges, this means that any user with an account in that system can log on to mysql as root by typing the above command. What is it that I should be doing to fix this?