How do I delete an existing event?

You can not delete an event, but you can choose to cancel the event:

Firstly visiting my My events in the top bar;
Then find the event you wish to cancel and click More then Cancel;
You will then be taken to a page to choose if you wish to refund fees or not;
You can write an optional message to all ticket buyers of your event. Once you confirm the cancellation an email will be automatically sent to all of your customers informing them of the cancellation.

The ticket price will be automatically refunded in all cases.

NOTE:

Only published events can be cancelled;
Once cancelled all orders will be automatically refunded minus the booking fee (unless you choose to pay for it);
Once you've cancelled your event, you can't make any changes, so make sure you make it non-public if you want the event removed from search engines such as Google.
When you cancel an event, the guest list will automatically disappear, so it might be a good idea to download a copy before you do so, in case you are contacted by a ticket buyer who has any questions.