The first volume of the The Authority was done by Warren Ellis and Bryan Hitch. This series introduced "a team of superheroes who promised to get the job done by whatever means necessary" called the Authority. The team consisted Jenny Sparks ("the Spirit of the 20th century"), Jack Hawksmoor ("Gods of the Cities"), Swift (a Tibetan woman with bird-like wings and talons), Apollo (a genetically engineered Superman pastiche who is gay), the Midnighter (a Batman pastiche and lover of Apollo), the Engineer (who replaced her blood with nano-technology) and Doctor (a Dutch junkie with magic powers). The team also had a living space ship called "The Carrier" This lasted 12 issues with 3 story arches. It was noted for it use for decompression (see my post on the topic) storytelling. Mark Miller and Frank Quitely (the latter uses a pen name) took over. They added a "revolutionary attitude" (the team challenging the status quo even when it comes to nations). Jenny dies (due to the 20th century ending) and Jack become leader. The US government form a counter team called G7 when they get annoyed by the Authority. Jenny Quantum (the spirit of the 21st century) was adopted by Apollo and Midnighter (who are now married). The Doctor overcame his addiction, but felt guilty for missing one the team's greatest challenge due to his addiction.
The second volume was written by Robbie Morrison and the art was mainly done by (a special related to the series feature other artists) Dwyane Turner. This lasted 15 issues (issue 0-14). This version was noted for "gratuitously placed and stylistically jarring four-letter" cuss words. In a four way crossover with Stormwatch: Team Achilles, Sleeper and Wildcats v3.0, the Authority took over the United States of America.
Volume 3 was actually called The Authority: Revolution. This was written by Ed Brubaker and had Dustin Nguyen and Richard Friend on art. This focused on the problems the team faces as the rulers of USA.
Volume 4 was the shortest volume. Grant Morrison was suppose to write a bimonthy series with Gene Ha on art duties. This lasted two issues. In this, a guy called Ken search for a downed submarine and finds not only the submarine, but also the Carrier. Ken later met the team. Keith Giffen was suppose to take over only for that to be scrapped.
Christos Cage and Darick Robertson wrote a miniseries called The Authority: Prime, which was basically volume 4.5. It focused on Stormwatch Prime and Authority teaming up to investigate a secret bunker.
Volume 5 had Dan Abnett and Andy Lanning writing and Simon Coleby as artist. This tied with the World's End crossover. Marc Bernardin and Adam Freeman took over as writers and Al Barrionuevo became artist. Christine Trelane (former co-leader of Stormwatch), Death Blow, former Stormwatch member Flint, Freefall from Gen13, Grifter from WildC.A.T.S., powerless ex-villain the High and Rainamaker also from Gen13 joined. Tom Talyor took over as writer and Mike S. Miller was the artist for the last two issues.
My source was the Wikipedia page "The Authority".