How do I know if a customer has paid?

You have several options for keeping track of your payment transactions.

Find all completed transaction immediately in your Mollie Dashboard on the Transactions page.
If you have linked Mollie to your webshop, you can use a webhook to process status updates.
You can view your transactions on the Mollie mobile app.
Receive text or email notifications for each transaction. In your Mollie Dashboard, go to Settings > Notifications.