A customer says they never got our email — how do I check if it was sent?

Open Settings → Web Settings → Email Delivery Logs. It lists every email your store has tried to send, with When, To, Subject, Type, the number of Recipients (Cc and Bcc included) and the Result.

Filter by result to narrow it down:

• Sent: the email left WCart. If the customer can't find it, ask them to check spam and confirm the address.
• Not sent — limit reached: your monthly allowance ran out. Buy more emails or upgrade on the Email Provider page.
• Not sent — no reCAPTCHA: a contact form email was held back because the form has no verified reCAPTCHA.
• Failed: the send didn't go through. A reason is shown under the result.

This is a different log from Inventory → Email Logs, which only covers back-in-stock alerts.