Uncap
Uncapping (上限解放, lit. "limit release" in Japanese) allows you to increase the maximum level of your characters, weapons, and summons. Doing this allows you to gain more stats and may unlock new effects or special abilities. To uncap something, either enter the Inventory page for the item and click the Uncap button, or click the Uncap tab from the Upgrade screen. Uncapping requires rupies in addition to either various materials or extra equipment, depending on what is being uncapped.
Uncapping Weapons
To uncap weapons, you either need an extra copy of the same weapon or a brick item. Seraphic Weapons are the main exception, requiring materials for every uncap. 
Red Steel Bricks and 
Steel Bricks (for R and SR weapons respectively) can be bought from the casino, but 
Gold Bricks can only be obtained through events, either by farming Gold Nuggets to trade in at the Shop, or as a very rare (0.1%) drop from a Nightmare-difficulty quest. Since Gold Bricks are required to recruit a Guild Wars character, it is highly recommended to uncap SSR gear the conventional way. You can also use 
Damascus Ingots from trading in moons or 
Damascus Crystals to upgrade SSR weapons, although this is not feasible until much later when it becomes easier to obtain duplicate characters or all of the crystals during an event. Weapons are best uncapped one at a time to enable final uncaps and make it easier to swap weapons if necessary.
Most equipment can be uncapped 3 times, increasing the maximum level by 15 for SR weapons and 20 for SSR weapons. Select SSR/SR weapons may have a 4th "blue" star. These uncaps use a large number of rare materials instead of extra weapons, but add an extra 50 levels (45 for SR) to the item, 5 more skill levels, and sometimes even a new weapon skill. Note that level and skill increases unlocked via a final uncap provide lesser benefits: A weapon going from level 99 to 100 may gain anywhere between 20-30 attack, yet that same weapon going to 101 will only gain roughly 7-8 attack. For most weapons, the third uncap increases the power of the weapon's Charge Attack, and the fourth uncap, if available, provides an even more powerful boost that will most likely make it the most damaging Charge Attack for your team.
Some of the SSR weapons with a blue star also require certain conditions to be met to unlock the star, most notably Omega weapons (defeating the respective Primal High Level in a hosted raid) and some character weapons (clearing the respective characters lv 45 fate episode).
Some weapons have a somewhat unusual structure of upgrading with benefits rather similar to or the same as uncapping (all of these are done through the Shop). Some of these involve upgrading SR rarity weapons into SSR equivalents, preserving uncap status, weapon and skill level while upgrading everything to SSR status including appropriate level cap (typically from 75 to 100) and upgraded charge attack, while also typically adding a new skill.
- Class Champion Weapons start as SR rarity weapons (called Replicas). A fully uncapped and leveled Replica can begin an upgrade process turning it into an SSR weapon followed by a minor stat boost step and the final step effectively giving the weapon a final uncap along with changing the weapons element.
- Revenant Weapons are initially uncapped normally, but most of their power comes from an extensive upgrade process which culminates in recruiting an Eternal.
- Bahamut Weapons are created from Rusted Weapons and Bahamut Horns (though they can also rarely drop directly), made SSR via the Nova upgrade and finally fully uncapped via the Coda upgrade.
- Seraphic Weapons cannot be uncapped by regular means, instead each uncap is performed through the shop with materials before these SR weapons are upgraded to SSR weapons.
- Xeno Weapons used to have an alternative upgrade in the shop that could not be performed on weapons that were given a final uncap normally. The regular uncap method was rarely used however and was removed as of November 2017 with the release of Xeno Cocytus Clash, making the shop upgrade the only means of performing a final uncap on this weapon series.
- Atma Weapons are created from relics, which in turn are created from fully uncapped and leveled Rusted Weapons, resulting in an SSR weapon of an element of the player's choice. They can be further upgraded into Ultima weapons.
Refer to the respective pages for further details.
Uncapping Summons
Summon uncaps are the same as weapon uncaps: use a copy of the summon, or use a stone from the casino (
Brightstone for R and 
Moonlight Stone for SR summons) or Unite and Fight (
Sunlight Stone for SSR summons). Events can usually reward enough summon copies to fully uncap, while Premium Draw summons will generally use stones since duplicate copies are more valuable reduced. When uncapping a Premium Draw summon with Sunlight Stones, it's best to wait until you have three stones to push the summon to 3★ instantly so you know exactly what you want to uncap.
Like weapons, most summons can be uncapped 3 times, with SRs gaining 15 more levels and SSRs 20 more levels per uncap. After 3 uncaps, the summon's aura will be upgraded, denoted by orange font. Certain summons may also have a blue star, which may require defeating the summon in a raid battle first, as well as a large number of rare materials. In addition to the extra 50 levels, these summons gain another upgrade to their aura (now denoted by purple font) and may also gain an upgraded call. These bonuses are obtained from uncapping, the actual level of the summon does not affect the performance of its call or aura.
In some cases, the blue star is unlocked by specific conditions, most notably summons with a raid which can be uncapped by hosting the respective raid boss.
Example: Apollo initially increases Light attack by 50% and grants an Attack boost when summoned. After three uncaps, the aura increases to 80%. It also has a blue star, which grants Light characters an additional 20% HP, and Apollo's call now additionally grants the Veil buff.
Some summons cannot be uncapped with stones. Arcarum Summons are another special case since they can only be obtained and uncapped through the associated Shop, with the final step turning the SR version into the respective SSR version.
Uncapping Characters
Players cannot have duplicate characters (see: Trade Moons), so uncapping characters requires various items to uncap instead. R characters can be uncapped 3 times, while most SR and SSR characters can be uncapped 4 times. Story SR characters begin with 3 stars and gain a 4th blue star after advancing far enough in the main quest, and some characters have a 5th star that grants multiple significant upgrades to the character. Unlocking that star requires leveling up to the normal maximum and clearing the fate episode that becomes available at that point.
Characters indirectly benefit a lot from being uncapped: the first uncap includes the level required for each character's Fate Episode skill, and the second-to-last normal uncap upgrades the Charge Attack and unlocks a new sprite and portrait art. Reaching certain other levels will upgrade skills, improving their effects and/or reducing their cooldowns. Character Level cap also increases by 10 extra levels per uncap and 20 for 5 star uncaps.
Character Uncap Materials
Items usually match the element of the character in question (e.g. uncapping Rackam requires fire materials and Colossus anima), but this is not always true (e.g. Rosetta needs Yggdrasil Anima).
Unlocking the Fifth ★
Characters that can be uncapped to a fifth blue ★ must first be uncapped to 4★ and at max level. This will unlock a new Fate Episode. Completing that episode will unlock the fifth ★, allowing the character to be uncapped. Note that after reaching max level again, another Fate Episode will unlock to upgrade another skill.
There are exceptions to this format, as some characters gain the ability to be uncapped to 5 ★ through story events (alongside a challenge battle rewarding an unique item used for the uncap itself). After such story events conclude, the uncap is available to all players while the uncap item can be traded for in the shop if needed.
Eternal characters have different, more complex requirements to unlock their 5 ★ as detailed here.
Story characters (and some Draw character versions based on them or other characters that appear in the story) with a 5 ★ uncap also additionally require progressing the main story to a certain chapter.