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 and some from Karla.

For more information about the differences between common spells and damage affinity check here.

Common Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Acid Surge	0	0	1	400	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.
Black Flame Arc	0	0	1	800	Self/Buff	Reinforce right weapon with flame, adding 80 Dark damage for 60 seconds. Ascended Effect: Increases Stamina Damage by 100%.	Firelink Shrine - Transposed from the Soul of Sister Friede.
Carthus Beacon	0	0	1	800	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	800	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.
Fecal Expulsion	0	0	1	300	Debuff/Ranged/AoE	Spew fecal matter from one's mouth, expelling 8 dung pies, inflicting 30 Toxic build up per bullet. Ascended Effect: Expel 12 dung pies.	Found in Irithyll of the Boreal Valley.
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.
Iron Flesh	0	0	1	600	Self/Buff	Increases Physical absorption by 75% and Aux. resistances by 50. Reduces Magic, Fire, Lightning and Dark absorption by 100%. Increases melee Deflection by 65 and slows movement. Lasts 15 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.
Poison Mist	0	0	1	200	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	600	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.
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.
Pungent Blade	0	0	1	800	Self/Buff/Utility	Imbue right-hand weapon with poison, adding 40 Poison to it for 60 seconds. Ascended Effect: Increases Poison effect to 80.	Firelink Shrine - Transposed from the Soul of the Rotted Greatwood.
Rapport	0	0	1	800	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	800	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.
Toxic Mist	0	0	1	300	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	600	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.

Physical Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Boulder Heave	0	0	1	320	Offensive/Ranged	Spew a boulder from one's mouth, dealing 500 Physical damage and 100 from the AoE. Ascended Effect: Increases Poise Damage and shoots further.	Firelink Shrine - Transposed from the Soul of a Stray Demon.
Earthfall	0	0	1	800	Offensive/Ranged/AOE	Call down boulders from the sky, dealing 300 Physical damage. 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	60	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.
Heave Earth	0	0	1	360	Offensive/Ranged	Heave earth from the ground, dealing 500 Physical damage. Ascended Effect: Costs 320 FP to cast.	Sold by Cornyx of the Great Swamp for 30,000 Souls after giving him the Profaned Pyromancy Tome.

Fire Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Bursting Fireball	0	0	1	150	Offensive/Ranged/AOE	Hurl a bursting fireball, dealing 60 Fire damage per bullet. 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	1	800	Offensive/Ranged/Homing/AOE	Release flaming chaos orbs from sky, dealing 300 Fire damage. Ascended Effect: Emits more orbs.	Firelink Shrine - Transposed from the Soul of the Demon Prince.
Chaos Bed Vestiges	0	0	1	350	Offensive/Ranged	Hurl chaos flame that scorches vicinity, dealing 500 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 700 Fire damage, and lava pools deal additional 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.
Fire Orb	0	0	1	180	Offensive/Ranged	Hurl giant fire orb, dealing 300 Fire damage. Ascended Effect: Gains improved tracking.	Sold by Cornyx of the Great Swamp for 3,000 Souls.
Fire Surge	0	0	1	120	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	100	Offensive	Sweep foes with fire whip, dealing 100 Fire damage per tick. Ascended Effect: Costs 60 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	120	Offensive/Ranged	Hurl fireball, dealing 150 Fire damage. Ascended Effect: Gains improved tracking.	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 600 Fire damage. Ascended Effect: Expands area of effect.	Sold by Cornyx of the Great Swamp for 15,000 Souls.
Flame Fan	0	0	1	120	Offensive/Utility	Make a large sweep with a fan of flame, dealing 150 Fire damage. Ascended Effect: Increases Poise Damage.	Sold by Cornyx of the Great Swamp for 5,000 Souls.
Floating Chaos	0	0	1	360	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	280	Offensive/Ranged	Hurl giant chaos fire orb, dealing 450 Fire damage, and the lava pool deals an additional 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	200	Offensive	Create powerful, giant flame in hand, dealing 500 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.
Immolation	0	0	1	200	Offensive/AOE	Immolate oneself, causing fire pillars to appear nearby, dealing 300 Fire damage to enemies and the player for 15 seconds. Also boosts Fire absorption by 50% for its duration. Ascended Effect: Extends duration to 30 seconds.	Firelink Shrine - Transposed from the Soul of the Old Demon King.
Lingering Chaos	0	0	1	180	Offensive/Ranged/Utility	Throw eight lingering chaos orbs that explodes on contact, dealing 100 Fire damage per bullet. Ascended Effect: Extends duration to 30 seconds.	Chaos Servants transposition in exchange for 5 Fire Seed and 5,000 Souls.
Profaned Flame	0	0	1	320	Offensive/Ranged	Engulf foes at range and burn them to ashes, dealing 500 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	200	Offensive	Flame burrows inside foes and ignites, dealing damage based on the Spell Tool used to cast it. Ascended Effect: Costs 100 FP to cast.	Sold by Cornyx of the Great Swamp for 20,000 Souls after giving the Quelana Pyromancy Tome to Karla.
Seething Chaos	0	0	1	260	Offensive/Ranged/Utility	Hurl chaos clump that seethes and explodes, dealing 300 (900) 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.

Dark Pyromancies

Spell					Effect Type	Effect Description	Acquisition
Black Fire Orb	0	0	1	280	Offensive/Ranged	Hurl a black fireball, dealing 350 Dark damage. Ascended Effect: Gains improved tracking.	Sold by Karla for 6,000 Souls after giving her the Grave Warden Pyromancy Tome.
Black Fire Scatter	0	0	1	400	Offensive/Ranged/AOE	Hurl an exploding black fireball, dealing 60 Dark damage per bullet. Ascended Effect: Gains improved tracking.	Sold by Grave Warden Lillian for 24,000 Souls.
Black Fire Vestiges	0	0	1	350	Offensive/Ranged/AOE	Hurl black fire flame that scorches vicitnity, dealing 500 (250) Dark damage. Ascended Effect: Gains much better tracking.	Firelink Shrine - Transposed from the Soul of High Lord Wolnir.
Black Fireball	0	0	1	120	Offensive/Ranged	Fire black fireball, dealing 100 Dark damage. Ascended Effect: Gains improved tracking.	Sold by Grave Warden Lillian for 3,000 Souls.
Black Flame	0	0	1	200	Offensive	Create giant, black flame in hand, dealing 500 Dark damage. Ascended Effect: Increases Poise Damage and uses push stagger animation.	Sold by Karla for 12,000 Souls after giving her the Grave Warden Pyromancy Tome.
Black Serpent	0	0	1	360	Offensive/Ranged/Homing	Release undulating black flame that runs along the ground, dealing 500 Dark damage. Ascended Effect: Expands the line length.	Firelink Shrine - Transposed from the Soul of High Lord Wolnir.
Dark Dance	0	0	1	650	Offensive/Ranged/Homing	Unleashes four large masses of Dark, dealing 200 Dark damage per bullet. Ascended Effect: Increases Poise Damage.	Sold by Karla for 15,000 Souls after giving her the Grave Warden Pyromancy Tome.
Mournful Flames	0	0	1	500	Offensive/Ranged/AOE	Hurl balls of black fire dealing 200 Dark damage, upon impact black flames seek foes, dealing 500 Dark damage. Ascended Effect: Emits 5 lines.	Firelink Shrine - Transposed from the Soul of High Lord Wolnir.
Recollection	0	0	1	360	Offensive/Ranged/Homing	Summons black fire-spitting orb, dealing 150 Dark damage per bullet. Ascended Effect: Emits more orbs.	Sold by Karla for 25,000 Souls after giving her the Grave Warden Pyromancy Tome.

Information here is accurate for version 1.76.