Some airlines may issue a new ticket or provide a flight credit voucher instead of offering a refund. 

A useful trick for normal circumstances: When they ask for your name for printing on the ticket, use your first initial instead of your full first name. (Many airlines now require your full first name, even if you purchase the ticket through a travel agent.) Thus if you can't use your "non-transferrable non-refundable" ticket, your spouse or some other member of your family might be able to.

[Airlines do not allow name changes on reservations and tickets, to prevent travel agents from buying up cheap fares with dummy names in anticipation of selling them to real people later. For this reason, it is important when traveling overseas to make sure you give your name to the travel agent exactly as it is listed on your passport. The name on your tickets must match the name on your passport, and once the reservations are made and the tickets issued, there's nothing you can do to correct the name. The travel agent will have to cancel the reservations and rebook the full itinerary using the correct name, assuming the seats are still available at that fare. If the tickets were non-refundable, there's nothing you can do.] 

Another trick is to have your travel agent talk to the airline, assuming you used him to purchase the ticket. Sometimes they will be able to swing a deal. 

Nontransferable tickets may still be useable by other people in your organization, if the address listed on the ticket was your business address, or if your organization's name appears on the ticket.

If the passenger named on the ticket died before the flight, most airlines will allow a name change, and many will even refund a nonrefundable ticket. 

Unused nonrefundable tickets can often be applied as a credit toward another nonrefundable ticket of equal or higher value on the same airline, less a small reticketing fee of $35 to $50. You'll also have to pay any difference in the fares, and you can do this for up to one year after the date of purchase. (Note that if the new ticket costs less than the old ticket, you probably won't be credited with the difference.) So if you bought a non-refundable ticket and decided to not use it, don't throw it away. If you only used part of the ticket (e.g., the originating flight but not the return), you probably can't do this; these policies are for completely unused tickets. 

If you lose your tickets, most airlines will require you to buy new tickets, and then issue a refund after 30 days, less a fee of $30 to $50. Thus 'losing' your tickets is not an effective means of getting a refund for non-refundable tickets.

Full fare tickets (e.g., unrestricted coach, business class, or first class), of course, usually allow changes without charge. 

Normally a reservation will cancel out automatically if you don't purchase a ticket within 24 hours. However, if you ask the airline to invoice you, usually this timeout is extended to 10 days to allow enough time for the invoice to reach you. You can still pay the invoice using a credit card. If you don't pay the invoice, the reservation cancels out at the end of the 10 day period. I was once able to get the 10 day period extended, but that was because the airline had transposed digits in my zip code, causing the invoice to arrive after the 10 day period had expired. 

Being hostile to airline personnel is the one sure way to prevent you from getting a refund or change. Be nice to them -- it doesn't hurt you, and probably will help you get what you want.