Is there a way to move e-mails from your folders in one account to a new e-mail account?

I have a new e-mail account but I have MANY saved e-mails in many folders with my old e-mail account. I need to know if there is a way to send them all to my new account, save them to my hard drive, or some other way not to lose them when I can no longer access that account. Thanks!
Is there a way to move e-mails from your folders in one account to a new e-mail account?
You will have to forward them one at a time to the new account.