Took a minute to do some research in the list of file names and extensions.  Found this info:

This indicates that a message that fails to send should be in the .PMX form and should be visible when reviewing queued mail but not editable.  It then follows that an empty "Review queued mail" list indicates all has been sent.

We never have problems sending emails, but this morning all of the emails I sent have come back as "Mail Delivery Failure".

Each returned email beings with:

Delivery has failed on the enclosed message for the following
reasons reported either by the mail delivery system on the mail
relay host or by the local TCP/IP transport module:

   *** TCP/IP error while processing job ***
   A network error has occurred while WinPMail attempted to send
   your message. This error could be caused by many conditions,
   including the following:
     * The remote host may have gone down.
     * Your network may have been broken or gone down.
     * You may have a configuration problem in your WINSOCK.DLL.
     * If running on a SLIP line, you may have experienced a
       timeout (increase the value in WinPMail's Network Config.
       Dialog), or the phone line may have gone down or suffered
       from excessive line noise.
   WinPMail will requeue your message and try again later.

Is this a message generated locally by Pegasus itself, or is it sent from a mail server?

Although it says WinPMail will requeue the messages and try again later, I have done "Review queued mail" and there are no messages in the queue so I don't know if they really are requeued.

I'm anxious to get these emails sent, but I can't tell where the problem is, whether I need to re-type them all again, or whether something is going to resend them automatically - and how I can tell that they really have been finally sent.

Can anyone offer any suggestions on what I should do?

Looks like Pegasus Mail could not connect to the SMTP server.  Perhaps it was a transient problem and Pegasus has since successfully sent the messages. 

Consider enabling internet session logging then sending a test message.  Review the resulting log file.

To enable internet session logging tick the "Create Internet session logs" option located on the General tab of Internet Mail Options (Tools > Internet Options).

Read the help file about internet session logging for details about what is logged and where to find the log files. 

Don't forget to turn logging off once done troubleshooting.

Thanks Brian. It seems that new emails are being sent OK now.

Just so I understand correctly - if the same error occurs again, does the failed email usually stay in Queued Mail and visible in File > Review Queued Mail ? And if the queued mail is empty then this means that Pegasus did eventually manage to send the email successfully?

[quote user="tiresias"]Just so I understand correctly - if the same error occurs again, does the failed email usually stay in Queued Mail and visible in File > Review Queued Mail ? And if the queued mail is empty then this means that Pegasus did eventually manage to send the email successfully?[/quote]

I assume so but don't know for sure.   Maybe someone will post with a more definite answer.