Killing the nest ranges from simply puffing some Ant and Wasp dust in a hole in the ground to the complex task of removing them from a building. If you are not confident about killing the nest yourself, request some help from an experienced person (seeing how they do it will help you get confidence) or ring a pest control company who will do the job for you for a significant fee.
Contrary to common wisdom, European wasps are not aggressive until they recognise your intent. The wasps returning to the nest feel good about what they are bringing home for the colony and the ones on the way out are blissfully unaware of what is coming until they get a face full of white powder. At that moment they loose their presence of mind and it takes 15 - 20 seconds for it to return and for them to ask "Who dunnit?" By then you need to be well clear unless you are well protected. For this reason you should not try to kill multiple nests at the same time - the ones from the first nest may be after you by the time you finish the second.
While petrol and diesel poured down the hole are effective they are not environmentally responsible. The most effective wasp killer I have found is Ant and Wasp dust. You can get it from a hardware store in a puffer pack. Bunnings have a Bayer product in a white 350 gram pack while Mitre 10 have the same product in a black pack, both about $20. The instructions say you may need to apply 125 - 250 g of product but I have found that 3 - 4 good puffs is enough - you should get around 10 nests out of a pack. The manufacturers data sheet warns it may cause irritation to the skin and to try to avoid inhaling it.
Approaching the nest:
Before you approach the nest, make sure you have cut the tip off the puffer pack. (The pack comes with a cap that can be put over the tip after use) Approach from the side, avoiding as much as possible the flight path. Put the nozzle in the hole and give three good puffs, four if it is a big nest. If you have to move grass aside make sure you wear gloves - this is the main way I have been stung.
Having delivered the goods, make your exit. Always make sure you have clear egress before attempting a kill.
Share your kill with the wasp patrol family so we can add it to the map and celebrate your victory.
Hard nests:
Hard nests are where access to the nest is difficult: maybe because of the terrain or the location of the nest in the middle of a blackberry bush. Make sure you are wearing maximum protection or try a different method. Baiting may be more appropriate. Is there anyone who could help me build a system that could deliver powder by compressed air down a long pipe? (I hate nests in blackberry bushes.)
Nests in houses present a challenge to both deliver the powder and remove the nest. Nests in ceilings ideally should be removed as the decaying larvae may cause both a smell and a stain but most often this is not practical