(Innate): Thresh does not gain armor per level. Instead, Thresh collects the souls of dead enemies by walking near. Souls permanently grant armor and ability power.
Champions and large minions always drop a harvestable soul. Small minions only sometimes drop a soul.
Damnation is a passive ability that causes enemy units to leave their souls behind when they die near Thresh. Regular minions and neutral monsters will sometimes leave a soul when they die. Enemy champions as well as large minions (siege minion or super minion) and large monsters will always leave a soul when they die. Thresh can pick up these souls by walking near them. Each soul gathered increases Thresh's armor, ability power, and the damage of Flay's passive.
The first soul will grant 1 bonus ability power and armor. Subsequent souls will grant 0.005 less, down to a minimum of 0.5 per stack. Souls cap at 999999. 
The 0.5 minimum is reached at 101 stacks, granting 75.75 bonus stats.
The maximum amount of ability power and armor gained is 500,024.75.
This amount of armor, including his base of 18, grants 99.98% damage reduction.
Even with the 35% armor penetration from Last Whisper, Thresh still benefits from 99.96% damage reduction.
Thresh surpasses 99% damage reduction at ~20,000 souls, rendering souls above this surplus with regards to armor.
It takes 49-94 souls to catch up to the armor levels of other level 18 champions (61 to 90 armor, including his base armor).
At 95, Thresh has the highest innate armor not including other champion's abilities.
It takes 146 souls to match a level 18 Taric with rank-5 Shatter - the highest armor available without items, runes and masteries (116 armor).
At 255 souls, Thresh gains a new buff, "Soul Gluttony". Soul Gluttony states that he has gained 255 souls as well as how much ability power and armor that is worth. The Damnation buff disappears, and starts again at 1 from your next soul. This will occur each time Thresh obtains 255 souls, with Soul Gluttony updating each time (e.g. 255, 510, 765, etc).
The full description of Damnation will list the total number of souls, armor and ability power. For example, Damnation may show "1" stack but will state "256" in the full description.
On Summoner's Rift, the drop rate for souls from small minions is 33%. However, the probability adjusts dynamically. If you are above or below your expected quota of soul drops from small minions, the probability lowers or rises by a small amount until you are back at your quota of drops 
It is worth stressing that this only takes into account souls dropped, and not souls collected.
The souls disappear after about 15 seconds.
Souls are visible to your allies at all times. If an enemy can see Thresh when the soul is spawned or at some point during the soul's existence, the soul becomes visible to the enemy team for the remainder of its existence.
Souls are colored differently depending on the team Thresh is on.
Souls look significantly smaller for both Thresh's allies and his enemies than they look for him
It is currently possible to collect souls while dead. It is unclear if this is a bug.
(Active): Thresh throws out his scythe, dealing magic damage to the first unit hit and pulling them toward him for 1.5 seconds.
Thresh can reactivate this ability to pull himself towards the bound enemy.
When activated, Death Sentence is a collision linear skillshot that deals magic damage to the first enemy unit it hits, stunning it for 1.5 seconds. During this time, the target is pulled a short distance towards Thresh twice, and Thresh may reactivate Death Sentence to dash towards the affected unit.
Death Sentence procs spell vamp and Rylai's slow similarly with all single target spells and abilities with full effect.
Death Sentence will damage Black Shield and its disabling effects will be negated.
On impact, Thresh will immediately pull the target for ~300 distance towards over 0.25 seconds. 0.5 seconds thereafter, it will again pull the target another ~300 distance towards him for over 0.25 seconds.
The target can be pulled past Thresh and through thin walls and temporary terrain if the target is close enough.
Death Sentence can be used to dash over walls if Thresh reactivates the ability to pull himself toward the target.
Thresh can move while pulling his target, changing the direction of the pull.
Thresh cannot autoattack until he either reactivates the ability or the duration of the grab ends.
If the target dies after the contact of the scythe (by either the initial magic damage or other source of damage), the scythe is immediately retrieved, that means Thresh can no longer pull himself to the target.
Even if Death Sentence's stun is removed (e.g. with Cleanse), the scythe is still attached on the target. That means Thresh can still pull himself to the target's position regardless and cannot autoattack until the scythe is retrieved.
Bot Champions will attempt to use Cleanse after being hit by the scythe.
If Thresh is inflicted with any ability-restricting crowd control effect after the scythe has hit a surviving enemy, Thresh cannot pull himself unless he removes the ailment on himself before the scythe is automatically retrieved.
(Active): Thresh throws his lantern towards the target location for up to 6 seconds. If an ally right-clicks it, they pick up the lantern and Thresh pulls them both back to him.
For the next 6 seconds, allies who come near the lantern (even while Thresh is holding it) gain a shield lasting 4 seconds that absorbs damage. Allies can only receive the shield once per cast. If Thresh moves more than ~1500 units away, the lantern will return to him.
Dark Passage is a ground targeted area of effect ability that places Thresh's lantern at the target location. Allied champions inside the area of effect gain an absorption shield that absorbs incoming damage for up to 6 seconds. Allied champions may only benefit from the shield once per cast. Additionally, allied champions can right-click on the lantern to grab it and dash toward Thresh.
The lantern provides vision in a small area for the entire duration.
Dark Passage can be used by allies to dash over walls.
Thresh cannot click on the lantern himself.
Dark Passage can be used to collect souls for Damnation. However its range in doing so is smaller than Thresh's range.
Thresh can break the leash range to instantly shield himself, if he has not been shielded by the ability already.
This will cause Thresh to practically "have" the lantern for the rest of the actual duration; this means, if the tether broke, Thresh can move to an allied champion and shield him, if the ally has not been shielded by the ability already
Allied champions can Teleport to Thresh's lantern. If the lantern's duration expires, the teleport will still continue to the lantern's previous location.
(Passive): Thresh's basic attacks deal bonus magic damage on each hit. The damage value is equal to the total number of souls collected plus a percentage of his attack damage, based on the amount of time since his last attack.
Maximum AD Ratio: 80 / 110 / 140 / 170 / 200 %
(Active): Thresh deals magic damage in a line beginning behind him. Enemies hit are pushed in the direction of the swing and then slowed for 1.5 seconds.
Flay passively causes Thresh's autoattacks to deal bonus magic damage equal to the number of gathered souls from Damnation plus a scaling percentage of his attack damage. The attack damage ratio increases over time while Thresh is not attacking.
The passive will accumulate some bonus damage during the wind-up of each attack.
This means that further away targets will take slightly more damage than closer targets in a persistent auto-attack fight, although for the most part the difference is less than 10 magic damage.
The time it takes for the passive damage to fully charge is currently ~10 seconds.
The percentage of his attack damage, based on the amount of time since his last attack is accumulated linearly.
This means that the % AD-based damage of his Flay passive does not benefit from attack speed.
Both the soul-based and attack damage-based damage of Flay are applied to the minor bolts from Runaan's Hurricane.
However, each time Thresh's autoattack or a bolt lands it resets the amount of time since Thresh's last auto attack. This means that the % AD-based damage of his Flay only adds a negligible amount of damage to the bolts from Runaan's Hurricane.
Flay's on-hit damage is not applied or consumed while attacking structures or wards.
The buff icon of Flay namely "Shackle" will appear on Thresh's status bar once he has learned this skill. The color of this icon indicates how much percentage the passive has charged, from green (>0%) to yellow to orange to red (100%).
When Thresh has successfully made an autoattack to the target, the icon will be briefly disappearing, denoting no bonus damage is granted by the passive for the time being.
When fully charged, Thresh scythe will visually glow in green. A distinctive gloomy slash sound effect can also be heard.
When activated, Flay is an point blank line area of effect ability that deals magic damage to enemy units in a linear area in front and behind Thresh. Enemy units hit will be knocked in the direction the ability was cast and they will be slowed for 1.5 seconds. For example, casting the ability behind Thresh will cause all enemies in front of him to be pulled toward him and enemies behind him will be pushed further back. Inversely, casting the ability in front of Thresh will cause enemies behind him to be pulled toward him and enemies in front of him to be pushed away.
Flay procs spell vamp and Rylai's slow similarly with all area of effect spells and abilities with diminished effect.
Flay will damage Black Shield and the slow and knockback/pull will be negated.
(Active): Thresh creates a prison of spectral walls around himself that remain active for 5 seconds. Enemy champions who walk through a wall take magic damage and are slowed by 99% for 2 seconds, but break that wall.
Once one wall is broken, the remaining walls' damage and slow duration is halved. An enemy can be afflicted by multiple walls.
The Box is a point blank ability that creates 5 stationary walls in a small area around Thresh that last up to 5 seconds. Enemy champions that walk through a wall take magic damage and are slowed for 2 seconds, but break the wall. After the first wall is broken, all remaining walls deal half the original damage when walked through and slow for 1 second.
Note that movement speed cannot be reduced below 110, and that slows that would reduce below 220 are modified. To put it another way: a reduction of 99% will not reduce you to 3 movement speed, but rather it will reduce you closer to 110 than a weaker slow would.