Marilith - Rank V
The Hunt to defeat Marilith becomes available immediately after you complete Hunt 08: A Scream from the Sky – Wyvern Lord (Rank II). Marilith is significantly harder to defeat, though. It becomes much easier later in the game, and you should attempt it after the events at the Stilshrine of Miriam and after Judge Bergan has been defeated at Mt Bur-Omisace.
Petitioner
The Tavernmaster in the Sandsea. He is standing behind the counter.
Mark Location
You can find Marilith in the Zertinan Caverns in an area called the Invitation to Heresy. You may or may not have had to venture into the Zertinan Caverns at this stage in the game, but, thankfully it is very easy to find the part of the caverns where Marilith Spawns.
Travel out into the Dalmasca Westersand from Rabanastre. Go through the Galtea Downs, south when you reach The Midfault, into the Simmering Horizons, and look for the cave in the southwest corner of the zone. This entrance leads directly into the section of the cavern that you need.
Marilith will appear after you have stayed in the area for 5 minutes or longer and will appear in the areas lit up with sunlight.
Hunt Battle
Level: 38
HP: 54,921
Steal: 
Fire Stone, Prime Tanned Hide, Fire Crystal
The difficult aspect of the fight against Marilith is the amount of damage that it can do to your party members. Make sure that you have a tank setup with 
Decoy active on that character at all times. You should also have either the Bubble spell or a 
Bubble Belt equipped to increase their overall HP.
Have one of your party members set up to keep 
Blind active on Marilith for the duration of the fight to reduce the amount of damage done. Try to keep all of your party members under the 
Haste status effect as well to counteract Marilith’s Slow spell. You also need to use 
Dispel on Marilith right off the bat to remove its Protect, Shell, and Haste status effects.
Marilith will start executing combos a lot more often as its HP gets low, so be prepared to do a fair bit of healing towards the end of the fight. Use Quickenings and Espers if you need to get in more damage.
Other tactics that you can use involve equipping your main tank with a 
Main Gauche and with 
Gauntlets to reduce the amount of damage that they take. Use 
Shielded Armor to ensure that Protect stays up the whole time.
Return to the Tavernmaster in Rabanastre once completed. The rewards for completing this Hunt include 2,200 gil, a Serpent Eye, and 3 Teleport Stones.
You should complete this Hunt during the intermission period in the Rabanastre following the events at the Stilshrine of Miriam.