Pyromancy

Pyromancies are on of the three available Spell types in Dark Souls III, the other ones being Sorceries and Miracles.
Casting pyromancies requires a Pyromancy Flame, as well as sufficient Intelligence and Faith.

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

List of Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Acid Surge	0	0	1	600	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: Lasts 12 seconds.	Sold by Cornyx of the Great Swamp for 12,000 Souls after giving him the Profaned Pyromancy Tome.
Boulder Heave	0	0	1	500	Offensive/Ranged	Spew a boulder from one's mouth, dealing 800 (250) Physical damage. Ascended Effect: Increases Poise Damage and shoots further.	Firelink Shrine - Transposed from the Soul of a Stray Demon.
Bursting Fireball	0	0	1	250	Offensive/Ranged/AOE	Hurl a bursting fireball, dealing 120 [100] Fire damage. Ascended Effect: Impacts leave lava pools.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Izalith Pyromancy Tome.
Carthus Beacon	0	0	1	500	Buff/Self	Boosts attack whilst attacking persists. Starts at 15%, increases to 30% with unbroken attacking. Lasts 30 seconds. Ascended Effect: Extends 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	1	500	Buff/Self	Reinforce right weapon with flame, adding 80 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.
Cataclysm	0	0	1	600	Offensive/Ranged/Homing/AOE	Release flaming chaos orbs from sky, dealing 800 Fire damage. Ascended Effect: Emits more orbs.	Firelink Shrine - Transposed from the Soul of the Demon Prince.
Chaos Bed Vestiges	0	0	1	500	Offensive/Ranged	Hurl chaos flame that scorches vicinity, dealing 800 [100] Fire damage. Ascended Effect: Increases Poise Damage.	Firelink Shrine - Transposed from the Soul of the Demon Prince.
Chaos Storm	0	0	1	75	Offensive/AOE	Erect multiple chaos fire pillars in vicinity, dealing 600 [350] Fire damage. Ascended Effect: Expands area of effect.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Izalith Pyromancy Tome.
Earthfall	0	0	1	600	Offensive/Ranged/AOE	Call down boulders from the sky, dealing 800 Physical damage per bullet. Ascended Effect: Emits more boulders.	Sold by Cornyx of the Great Swamp for 60,000 Souls after giving him the Profaned Pyromancy Tome.
Earthquake	0	0	1	75	Offensive/AOE	Wretch earth in multiple locations in vicinity, dealing 600 Physical damage. Ascended Effect: Expands area of effect.	Sold by Cornyx of the Great Swamp for 15,000 Souls after giving him the Profaned Pyromancy Tome.
Fire Orb	0	0	1	250	Offensive/Ranged	Hurl giant fire orb, dealing 400 (200) Fire damage. Ascended Effect: Impact leaves lava pool.	Sold by Cornyx of the Great Swamp for 3,000 Souls.
Fire Surge	0	0	1	50	Offensive	Emit a constant stream of fire, dealing 150 Fire damage. Ascended Effect: Hits more often.	Sold by Cornyx of the Great Swamp for 1,000 Souls after giving him the Great Swamp Pyromancy Tome.
Fire Whip	0	0	1	50	Offensive	Sweep foes with fire whip, dealing 100 Fire damage per tick. Ascended Effect: Costs 25 FP per tick.	Sold by Cornyx of the Great Swamp for 6,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Fireball	0	0	1	100	Offensive/Ranged	Hurl fireball, dealing 300 (200) Fire damage. Ascended Effect: Impact leaves lava pool.	Sold by Cornyx of the Great Swamp for 1,000 Souls.
Firestorm	0	0	1	50	Offensive/AOE	Erect multiple fire pillars in vicinity, dealing 500 Fire damage. Ascended Effect: Expands area of effect.	Sold by Cornyx of the Great Swamp for 15,000 Souls.
Flame Fan	0	0	1	100	Offensive/Utility	Make a large sweep with a fan of flame, dealing 200 Fire damage. Ascended Effect: Costs 50 FP to cast.	Sold by Cornyx of the Great Swamp for 5,000 Souls.
Flash Sweat	0	0	1	500	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.
Floating Chaos	0	0	1	500	Offensive/Ranged/Homing	Summons fire-spitting chaos orb, dealing 150 Fire damage per bullet. Ascended Effect: Emits more fireballs.	Sold by Cornyx of the Great Swamp for 25,000 Souls after giving him the Izalith Pyromancy Tome.
Great Chaos Fire Orb	0	0	1	400	Offensive/Ranged	Hurl giant chaos fire orb, dealing 500 [100] [350] Fire damage. Ascended Effect: Extends the duration of the lava pool to 8 seconds.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Izalith Pyromancy Tome.
Great Combustion	0	0	1	400	Offensive	Create powerful, giant flame in hand, dealing 600 [200] Fire damage. Ascended Effect: Deals more Poise Damage and causes stagger to floor animation.	Sold by Cornyx of the Great Swamp for 3,000 Souls after giving him the Great Swamp Pyromancy Tome.
Heave Earth	0	0	1	500	Offensive/Ranged	Heave earth from the ground, dealing 600 Physical damage. Ascended Effect: Costs 400 FP to cast.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Profaned Pyromancy Tome.
Immolation	0	0	1	400	Offensive/AOE	Immolate oneself, causing 100 Fire damage to nearby foes and burning 1% HP every second for 15 seconds. Ascended Effect: Burns oneself for 0.25% damage.	Firelink Shrine - Transposed from the Soul of the Old Demon King.
Iron Flesh	0	0	1	500	Self/Buff	Increases Physical absorption by 40% and Aux. resistances by 50. Reduces Lightning absorption by 60%. Increases melee Deflection by 65 and slows movement. Lasts 17 seconds. Ascended Effect: Extends duration to 30 seconds.	Sold by Cornyx of the Great Swamp for 6,000 Souls after giving him the Carthus Pyromancy Tome.
Lingering Chaos	0	0	1	75	Offensive/Ranged/Utility	Throw lingering chaos orb that explodes on contact, dealing 400 Fire damage. Ascended Effect: Extends duration to 20 seconds.	Chaos Servants transposition in exchange for 5 Fire Seed and 100 Souls.
Poison Mist	0	0	1	400	Debuff/Utility/AOE/Ranged	Summons patch of poison that lasts 6 seconds and deals 25 Poison build up per tick. Ascended Effect: Lasts 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	1	500	Self/Buff/Debuff	Increases damage by 20% and Stamina Recovery by 30 for 30 seconds. Lose 1% HP per second whilst active. Ascended Effect: Lose 0.25% per second.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Profaned Flame	0	0	1	300	Offensive/Ranged	Engulf foes at range and burn them to ashes, dealing 400 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.
Profuse Sweat	0	0	1	500	Self/Buff	Increases Poison, Toxic, Bleed, Frost and Curse resistance by 200 for 60 seconds. Ascended Effect: Increases resistances by 400.	Sold by Cornyx of the Great Swamp for 2,000 Souls.
Rapport	0	0	1	1,000	Debuff/Ranged	Charm the enemy, making them a temporary ally for 30 seconds. Ascended Effect: Extends duration to 60 seconds and boosts converted enemy power by 20%.	Sold by Cornyx of the Great Swamp for 50,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Rockflesh	0	0	1	500	Self/Buff	Grants rockflesh, increases absorption by 30% and prevents staggering, but slows movement for 30 seconds. Ascended Effect: Extends duration to 60 seconds.	Firelink Shrine - Transposed from the Soul of the Old Demon King.
Sacred Flame	0	0	1	300	Offensive	Flame burrows inside foes and ignites, dealing Fire damage based on weapon damage. Ascended Effect: Costs 250 FP to cast.	Found in Smouldering Lake.
Seething Chaos	0	0	1	400	Offensive/Ranged/Utility	Hurl chaos clump that seethes and explodes, dealing 300 (600) Fire damage. Ascended Effect: Increases Poise Damage.	Sold by Cornyx of the Great Swamp for 5,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Toxic Mist	0	0	1	500	Debuff/Utility/AOE/Ranged	Summons patch of poison that lasts 6 seconds and deals 10 Toxic build up per tick. Ascended Effect: Lasts 12 seconds.	Sold by Cornyx of the Great Swamp for 4,000 Souls after giving him the Carthus Pyromancy Tome.
Warmth	0	0	1	750	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.

Information here is accurate for version 1.71.