Reach the rank of Chief Sergeant in your Grand Company, to have access to Grand Company leves in Mor Dhona. This requires clearing rank 1 of the Grand Company's hunting log, as well as clearing Dzemael Darkhold.
You will need to unlock the following non-MSQ dungeons:
Speak with Jalzahn with your Atma Zodiac Weapon equipped who will explain that in order to further enhance your weapon you need to recreate the valorous deeds of the Zodiac Braves, who were revered heroes in Thavnairian legend. These deeds are recorded in a collection of nine sacred texts called the “Trials of the Braves”. He will send you to Rowena in Revenant's Toll - who in turn will direct you to G'jusana who is nearby. After speaking to G'jusana, the quest will move to your "completed quest log".
G'jusana will sell you one book at a time for 100 Allagan Tomestone of Poetics each. Each book contains a series of tasks to complete: 10 sets of specific monsters to kill and 3 dungeons, 3 FATEs and 3 levequests to complete. Once all the tasks are completed in one book, return to G'jusana to buy the next one. You must finish all the tasks in one book before completing the next - however you can complete the books in any order. Each book will improve a stat on your Atma Zodiac Weapon.
Each book belong to one of four categories. Each category improves a stat on your Atma Zodiac Weapon and each category is called a different name depending on the job. However, the books themselves and the tasks inside them remain the same regardless of your job.
You can only ever own one book at a time. Each book is specific to a weapon and you must be equipped with the Atma Zodiac Weapon you wish to enhance when making the purchase.
Must equip the corresponding weapon in order for any completed task to be registered.
Buying a second book will automatically dispose of the first one. If the first book has not been completed it will need to be redone.
When working on your second weapon onwards, it is highly recommended to do the books in a specific order in parallel with the "Zodiac Braves" subquests, as the two steps have a lot of overlap in required dungeons. This recommended book order is:
Fall I → Earth I → Wind I → Fall II → Fire I → Fire II → Wind II → Netherfire I → Netherfall I
You can achieve bronze, and can even be dead, and still have the FATE count.
Each map has a maximum number of FATEs available at any one time. Therefore you may need to clear other FATEs in order to get the one you need to spawn.
Also, FATEs generally spawn in such a way that their blue circular areas don't overlap. Therefore, if a different FATE has spawned in the general area of the one you need, clear it out as soon as possible.
Some FATEs will only spawn when triggered (either by talking to an NPC at the location or in a nearby location; or after the completion of a previous FATE as part of a chain). Examples:
Note that only the imperial soldiers and the dogs count toward completion of The Enemy of My Enemy - the sylphs will be aggro'd to the imperial enemies, so avoiding the sylphs and letting them attack can allow for clearing the fate slightly faster.
The North Tidegate FATEs seem to spawn the South Tidegate FATEs soon after completion, and vice versa. Therefore, if the wrong FATE chain is up, complete it and the other will probably spawn.
Note that completion is tied to the Kobold Toolboxes (crates), and then the boss 59th Order Matriarch Go Zu that spawns after the crates are destroyed. The regular kobolds are not required to complete the fate. Following the NPCs and keeping them alive can speed up completion a bit.
Surprise at Zelma's Run, Upper La Noscea (X:26.0, Y:18.0) - The FATE will automatically fail about a minute after spawning if no players are present, as the NPCs will be killed by the enemies. This FATE can be anticipated shortly before it spawns, as two Marauders (a male Roegadyn and a female Miqo'te) with Chocobos will spawn near Camp Bronze Lake and march towards the location of the FATE.
Air Supply at Ixali Logging Grounds, North Shroud (X:19.0, Y:20.0) - Only the Airstones (crates) and Ixali Swiftbeaks (which spawn when crates are attacked) will count towards completion. The other Ixali enemies and the dogs will not count. The FATE spawns with some NPCs that will attack the crates, so some of the Swiftbeaks may already be spawned when you arrive.
It is generally recommended to seek out the FATEs first among all tasks in the book, since you may need to wait for them to spawn. Use the other book tasks as time-killers in the meantime.
You can visit other worlds on your data center through the World Visit System, or visit other data centers through Data Center Travel, to check if the FATEs you need have spawned there. Note that visiting another world requires going to Ul'dah, Limsa Lominsa, or New Gridania and interacting with the aetheryte, while visiting another data center requires logging out and then right-clicking your character's name in the character select menu.
You must have your Atma Zodiac Weapon equipped for the final boss for it to count.[1] You can equip a stronger high-level weapon to make the dungeon go faster, but DON'T FORGET to re-equip your Atma weapon prior to engaging and killing the final boss, or else you'll have to re-run the entire dungeon to get credit.
You must be alive when the final boss of the dungeon is killed. If you are dead, it will not count towards completion of the book.[1]
General levequests and Grand Company levequests are given out by two different NPCs at each location. Make sure you are talking to the correct one.
Only a limited number of levequests are offered at any one time. You may need to complete one of the levequests in order to reveal the one you need.
If you're low on allowances, completing a levequest without turning it in for a reward will refresh the list of available quests. You can also check other areas for any of the leves you need prior to completing the others you don't. Completing a levequest in any zone will reset the available leves everywhere.
If you have a lot of allowances, holding onto several leves at once can help ensure you have access to the correct one. Levemetes offer leves from a set pool per area, but they cannot offer a leve you are currently holding.
It is possible to hold more leves than a levemete offers at a time - for example, the Northern Thanalan levemete has a pool of 5 leves, offering 4 of them at a time, but it is possible for you to hold all 5 of his leves at once. Whenever the available leves are reset (by completing a leve from any levemete, or at daily reset), this levemete will randomly pick 4 leves from his pool to offer. If a leve in the final list is one you are already holding, then it will be hidden. This means that, if you collect all 4 of this levemete's leves, reset the leve list, and then come back, there is a chance that he has the 5th leve on offer for you to collect.
The leves required per book per area are as follows:
Levequests are only counted as "completed" when you collect the reward from the Levemete, and this is the only time where the game checks whether your Atma weapon is equipped. As such, it is possible to do the leve with another weapon (or another class altogether) and switch back to your Atma before collecting the rewards, and it will still count.
Tips for specific leve types:
For the "Necrologos" leves - Only the one in Coerthas Central Highlands requires killing enemies (and even then, they immediately respawn, so you can simply kill the same enemy repeatedly). For the leves in Mor Dhona and Northern Thanalan, the enemies cannot drop pages, so they can be ignored - just pick up the pages on the ground and kill enemies that spawn from the pages.
For the "kill enemies for 5 minutes" leves - there are generally 2 common enemies that spawn in pairs together from the start of the leve and a rare enemy that will only start spawning paired with one of the common enemies starting exactly 1 minute after the leve has initiated, even if you haven't killed any of the common enemies up to that point. After at least one of each of the 3 types of enemies have been killed, you are not required to linger around the leve area - as long as you stay in the zone, you are free to fly around and kill enemies for other book objectives. You cannot level sync for FATEs while the leve is running, however.
For "find the impostors" leves - you an skip the process of poking an enemy for an item, using the item to reveal impostors, and then killing the impostors - simply kill the enemies without checking, as impostors will be revealed after taking one hit of damage. There is no penalty for killing non-impostors.
For enemies, click the book item links above for specific coordinates.
Closing notes:
Your last book will still remain in your Key Item Inventory after completion. To remove it, simply return to G'jusana and select the Disposal option. Alternatively, drag it out of your Key Items inventory to discard it like any other item.
This is also a good opportunity to train your Chocobo if you wish.
900 Allagan Tomestone of Poetics (100 at a time)
levequests in:
Leves of Camp Bluefog)
Dzemael Darkhold.
Book of Skyfire I
FATEs
The Enmity of My Enemy - Requires 
Breaching North Tidegate - Requires 
The Ceruleum Road - Triggered by speaking to the Wary Merchant at Camp Bluefog, Northern Thanalan (x21,y29).
Taking It to the Stars
