The Advent's weapon is in the Civilian research tree, at the end of the Culture research path - it's called the Deliverance Engine.
What it does is throw a huge chunk of your culture towards any planet, friendly or hostile. This serves several purposes:
1) The Advent have research that displays any grav well where they have culture, so you can peek anywhere on the map for a fairly long while, since it takes some time for the culture effect to wear off.
2) The Advent's culture techs give all of their ships extra shield mitigation when they're in friendly culture. The other races have their own bonuses. So, shooting the Engine at a planet you're about to invade means the defending fleet doesn't get their bonus and you get yours. This can make a big difference in a fight with about even fleets.
The Vasari superweapon is the Kotsura Cannon, it damages and disables planetary structures for 3 minutes, and damages and disables ships for I think 15 seconds. It's also good for busting through heavily defended planets, since shutting down all the repair bays for 3 minutes and dealing some damage to everything isn't a bad thing.