Q: I have received an invoice for something I have never bought

This is an emailed receipt (iTunes doesn't send invoices) ? If it is you are sure that it was a genuine email from Apple and not a phishing attempt trying to get your account and payment details ?

Phishing and similar emails : Phishing & Other Suspicious Emails

Phishing emails : Identifying fraudulent "phishing" email - Apple Support

Genuine iTunes emails : Identifying legitimate emails from the iTunes Store - Apple Support

Viewing your account's purchase history : See your purchase history in the iTunes Store - Apple Support (on an iOS device you can view the last 90 days purchases via http://reportaproblem.apple.com)

If it's a genuine email and it shows on your account's purchase history, and nobody else has access to your account or devices/computers, then contact iTunes Support : http://www.apple.com/support/itunes/contact/  . And change your account's password http://appleid.apple.com - after changing it : What to do after you change your Apple ID email address or password - Apple Support.

If it is a phishing email forward it to Apple : reportphishing@apple.com , and then delete it.

If it's a phishing attempt and you clicked on the link in the email and entered your account id and password then change your password immediately