It seems from several forums that quite a few people are having starting problems with various audi models.
If your symptom is that the engine fires up and then dies almost immediately, over and over again, then you probably haven't got fuel pump or ignition trouble, you have immobiliser trouble. What is going on is that the immobiliser in the car is not recognising your key. This can happen at random - one start, all is ok, next time, no start.
It has recently happened to my 1998 A6.
My local audi dealer wasn't much help, offered me an appointment for the following week and no advice.

What should happen in my A6 is that when switching on the ignition, the key symbol on the dash should light momentarily and then go out when the key is recognised. If it fails to be recognised, the light starts flashing and the car won't start.

There are two things you can do:
1. Use the workshop key, which while it has no remote locking, doesn't need to be recognised
2. If you're stuck, try leaving the ignition turned on for about 10/15 minutes. Twice now this has worked for me - eventually the key is recognised, the flashing light goes out and you're on your way.

Now why couldn't my audi dealer tell me that?
If anyone has had this fault corrected by a dealer, I'd be glad to know how long it took them and what it cost.

 

1. All vehicles keys need to be programed to the imobilizer. Including valet and emergency keys.
2. There is no programing for imobilizer that happens when the key is turned on for extended periods of time.