First things first, make sure extensions are enabled. This is more of an issue for ExpressionEngine 2.0, but it's a good first step all the same.
Next, make sure you've added the tag ({exp:subscriber:form form_id="1"}) to the form and also make sure that you see the hidden input in the rendered form. While you're here, verify that the form_id is correct, there's nothing more frustrating than realizing you had your subscribers going to the wrong list.
Next, you should check the logs. When there's an issue with the MailChimp and Campaign Monitor APIs, they're logged here. Sometimes the fix is as simple as correcting the API key or list ID.
If using Freeform, make sure the fields are represented within the Freeform Control Panel. If they aren't there, Subscriber does not receive them.
If everything looks good so far, you should check with MailChimp and Campaign Monitor's status pages, because they may be having issues at the moment. Typically in these cases the email addresses are added to the list, but there may be a delay. Also, if you're using double opt-in, check the subscribing email address for confirmation emails.
If your issue isn't fixed, feel free to ask a question in the support forum. If you're having some serious issues, be prepared to email me your CP and FTP credentials, and possibly your MailChimp or Campaign Monitor credentials.