Pyromancy

Pyromancies are on of the four available Spell types in Dark Souls III, the other ones being Sorceries, Miracles and Hexes.
Casting pyromancies requires a Pyromancy Tool.

Most pyromancies can be learned from the spell trainer Cornyx of the Great Swamp.

For more information about spells check here.

Common Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Acid Surge	0	0	1	30	Debuff/Utility/AOE/Ranged	Emit acid cloud that lasts 6 seconds and corrodes enemies, dealing 5 + 0.05% HP damage and reducing 1 Durability per tick. Ascended Effect: increases duration to 12 seconds.	Sold by Cornyx of the Great Swamp for 12,000 Souls after giving him the Profaned Pyromancy Tome.
Carthus Beacon	0	0	1	40	Buff/Self	Boosts attack whilst attacking persists. Starts at 15%, increases to 30% with unbroken attacking. Lasts 30 seconds. Ascended Effect: increases duration to 60 seconds.	Sold by Cornyx of the Great Swamp for 8,000 Souls after giving him the Carthus Pyromancy Tome.
Carthus Flame Arc	0	0	2	50	Buff/Self	Reinforce right weapon with flame, adding 75 Fire damage for 60 seconds. Ascended Effect: increases Stamina Damage by 100%.	Sold by Cornyx of the Great Swamp for 10,000 Souls after giving him the Carthus Pyromancy Tome.
Fecal Expulsion	0	0	1	15	Debuff/Ranged/AoE	Spew fecal matter from one's mouth, expelling 8 dung pies, inflicting 30 Toxic build up per bullet. Ascended Effect: emit an additional four projectiles.	Found in Irithyll of the Boreal Valley.
Flash Sweat	0	0	1	30	Self/Buff	Intense sweating increases fire damage absorption. Ascended Effect: increases Fire absorption by 50%.	Sold by Cornyx of the Great Swamp for 3,000 Souls after giving him the Great Swamp Pyromancy Tome.
Iron Flesh	0	0	2	30	Self/Buff	Increases Physical absorption by 75% and Aux. resistances by 45. Reduces Magic, Fire, Lightning and Dark absorption by 100%. Increases melee Deflection by 65 and slows movement. Lasts 15 seconds. Ascended Effect: increases duration to 30 seconds.	Sold by Cornyx of the Great Swamp for 6,000 Souls after giving him the Carthus Pyromancy Tome.
Poison Mist	0	0	1	20	Debuff/Utility/AOE/Ranged	Summons patch of poison that lasts 6 seconds and deals 30 Poison build up per tick. Ascended Effect: increases duration to 12 seconds.	Sold by Cornyx of the Great Swamp for 2,000 Souls after giving him the Great Swamp Pyromancy Tome.
Power Within	0	0	2	50	Self/Buff/Debuff	Increases damage by 20% and Stamina Recovery by 30 for 30 seconds. Lose 1% HP per second whilst active. Ascended Effect: reduces the HP loss to 0.5% HP per second.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Profuse Sweat	0	0	1	25	Self/Buff	Increases Poison, Toxic, Bleed, Frost and Curse resistance by 200 for 60 seconds. Ascended Effect: increases resistance boost to 400.	Sold by Cornyx of the Great Swamp for 2,000 Souls.
Rapport	0	0	2	100	Debuff/Ranged	Charm the enemy, making them a temporary ally for 30 seconds. Ascended Effect: increases duration to 60 seconds and boosts allied enemy damage by 20%.	Sold by Cornyx of the Great Swamp for 50,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Rockflesh	0	0	2	50	Self/Buff	Grants rockflesh, increases absorption by 30% and prevents staggering, but slows movement for 30 seconds. Ascended Effect: increases duration to 60 seconds.	Firelink Shrine - Transposed from the Soul of the Old Demon King.
Toxic Mist	0	0	1	25	Debuff/Utility/AOE/Ranged	Summons patch of poison that lasts 6 seconds and deals 15 Toxic build up per tick. Ascended Effect: increases duration to 12 seconds.	Sold by Cornyx of the Great Swamp for 4,000 Souls after giving him the Carthus Pyromancy Tome.
Warmth	0	0	2	100	Buff/AOE	Summons healing orb that heals self, allies and foes for 30 HP per second. Lasts 60 seconds. Ascended Effect: only affects self and allies.	Mound Makers transposition in exchange for 15 Vertebra Shackle and 100 Souls.

Fire Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Bursting Fireball	0	0	1	15	Offensive/Ranged/AOE	Hurl a bursting fireball, dealing 200 Fire damage. Ascended Effect: emits an additional fireball.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Izalith Pyromancy Tome.
Cataclysm	0	0	2	50	Offensive/Ranged/Homing/AOE	Release flaming chaos orbs from sky, dealing 200 Fire damage. Ascended Effect: increases emitted projectile rate to 4 per second.	Firelink Shrine - Transposed from the Soul of the Demon Prince.
Chaos Bed Vestiges	0	0	2	30	Offensive/Ranged	Hurl chaos flame that scorches vicinity, dealing 400 Fire damage (100 for reduced AoE). Ascended Effect: increases projectile speed to 38 m/s.	Firelink Shrine - Transposed from the Soul of the Demon Prince.
Chaos Storm	0	0	2	10	Offensive/AOE	Erect multiple chaos fire pillars in vicinity, dealing 400 Fire damage, and lava pools deal additional 150 Fire damage. Ascended Effect: increases emitter radius to 30 meters.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Izalith Pyromancy Tome.
Fire Orb	0	0	1	15	Offensive/Ranged	Hurl giant fire orb, dealing 300 Fire damage (100 for reduced AoE). Ascended Effect: increases projectile homing to 40 degrees.	Sold by Cornyx of the Great Swamp for 3,000 Souls.
Fire Surge	0	0	1	5	Offensive	Emit a constant stream of fire, dealing 80 Fire damage. Ascended Effect: reduces hit cooldown to 0.3 seconds.	Sold by Cornyx of the Great Swamp for 1,000 Souls after giving him the Great Swamp Pyromancy Tome.
Fire Whip	0	0	1	4	Offensive	Sweep foes with fire whip, dealing 100 Fire damage per tick. Ascended Effect: reduces hit cooldown to 0.05 seconds.	Sold by Cornyx of the Great Swamp for 6,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Fireball	0	0	1	10	Offensive/Ranged	Hurl fireball, dealing 200 Fire damage (50 for reduced AoE). Ascended Effect: increases projectile speed to 30 m/s.	Sold by Cornyx of the Great Swamp for 1,000 Souls.
Firestorm	0	0	1	5	Offensive/AOE	Erect multiple fire pillars in vicinity, dealing 300 Fire damage. Ascended Effect: increases emitter radius to 30 meters.	Sold by Cornyx of the Great Swamp for 15,000 Souls.
Flame Fan	0	0	1	10	Offensive/Utility	Make a large sweep with a fan of flame, dealing 150 Fire damage. Ascended Effect: increases Poise Damage to 20.	Sold by Cornyx of the Great Swamp for 5,000 Souls.
Floating Chaos	0	0	2	30	Offensive/Ranged/Homing	Summons fire-spitting chaos orb, dealing 80 Fire damage per bullet. Ascended Effect: increases emitted projectile rate to 10 per second.	Sold by Cornyx of the Great Swamp for 25,000 Souls after giving him the Izalith Pyromancy Tome.
Great Chaos Fire Orb	0	0	2	25	Offensive/Ranged	Hurl giant chaos fire orb, dealing 400 Fire damage (100 for reduced AoE), and the lava pool deals an additional 150 Fire damage. Ascended Effect: reduces lava hit cooldown to 0.5 seconds.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Izalith Pyromancy Tome.
Great Combustion	0	0	1	25	Offensive	Create powerful, giant flame in hand, dealing 350 Fire damage (100 for reduced AoE). Ascended Effect: increases Poise Damage to 50.	Sold by Cornyx of the Great Swamp for 3,000 Souls after giving him the Great Swamp Pyromancy Tome.
Immolation	0	0	1	10	Offensive/AOE	Immolate oneself, causing fire pillars to appear nearby, dealing 200 Fire damage to enemies and the player for 15 seconds. Also boosts Fire absorption by 50% for its duration. Ascended Effect: increases duration to 30 seconds.	Firelink Shrine - Transposed from the Soul of the Old Demon King.
Lingering Chaos	0	0	1	15	Offensive/Ranged/Utility	Throws a lingering chaos orb that explodes on contact, dealing 200 Fire damage. Ascended Effect: increases duration to 30 seconds.	Chaos Servants transposition in exchange for 5 Fire Seed and 5,000 Souls.
Profaned Flame	0	0	1	25	Offensive/Ranged	Engulf foes at range and burn them to ashes, dealing 350 Fire damage. Ascended Effect: emits a secondary flame.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Profaned Pyromancy Tome.
Sacred Flame	0	0	1	15	Offensive	Flame burrows inside foes and ignites, dealing damage based on the Spell Tool used to cast it. Ascended Effect: none.	Sold by Cornyx of the Great Swamp for 20,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Seething Chaos	0	0	1	25	Offensive/Ranged/Utility	Hurl chaos clump that seethes and explodes, dealing 200 (400) Fire damage. Ascended Effect: increases Poise Damage to 100.	Sold by Cornyx of the Great Swamp for 5,000 Souls after giving the Quelana Pyromancy Tome to Karla.

Physical Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Boulder Heave	0	0	2	25	Offensive/Ranged	Spew a boulder from one's mouth, dealing 300 Physical damage and 100 from the AoE. Ascended Effect: increases projectile speed to 16 m/s.	Firelink Shrine - Transposed from the Soul of a Stray Demon.
Earthfall	0	0	2	50	Offensive/Ranged/AOE	Call down boulders from the sky, dealing 200 Physical damage. Ascended Effect: increases emitted projectile rate to 4 per second.	Sold by Cornyx of the Great Swamp for 60,000 Souls after giving him the Profaned Pyromancy Tome.
Earthquake	0	0	2	5	Offensive/AOE	Wretch earth in multiple locations in vicinity, dealing 300 Physical damage. Ascended Effect: increases emitter radius to 25 meters.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Profaned Pyromancy Tome.
Heave Earth	0	0	1	25	Offensive/Ranged	Heave earth from the ground, dealing 350 Physical damage. Ascended Effect: emits two additional lines.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Profaned Pyromancy Tome.

Information here is accurate for version 1.91.