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So I know how to calculate spell DC (8 + proficiency + modifier) and spell attack rolls (1d20 + proficiency + modifier), but I am not clear on spell damage.
Namely, do you add your spell modifier (e.g. int for wizards) to spell damage?
For example, burning hands does 3d6 fire damage or half as much if the opponent succeeds on a dex save.
If you have a Wizard with 16 Int casting it with a 1st-level spell slot, does the spell do 3d6 (or half) damage, or 3d6+3?
No
A spell does the damage it says it does. In the case of Burning Hands, it says it does 3d6, so it does 3d6.
There are a few exceptions to this: the school of Evocation for Wizards allows you to add your spellcasting modifier to the damage of evocation spells, and certain Cleric domains allow you to add your spellcasting modifier to the damage dealt by cantrips. There is an invocation that allows a Warlock to add their spellcasting modifier to the damage dealt by Eldritch Blast. The Dragonblood Sorcerer can add their spellcasting modifier to spells that deal elemental damage matching their bloodline type.
Outside of these, though, the only reason a spell would add your spellcasting modifier to damage is if it says that it does. I'm not aware of any spells that currently do this.
In the chapter 9 of the player's handbook, there is a section titled "Damage Rolls", which covers the general rules for dealing damage with weapons or spells.
Damage Rolls
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any m odifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.
When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage. A spell tells you which dice to roll for damage and whether to add any modifiers.
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell’s damage is rolled once for all creatures caught in the blast.
According to this rule, the spell description should tell you if the spell's damage requires adding an ability modifier. Different spells do different damage, and not all of them add a modifier to the attack.
In most cases, however, spells do not require the addition of your spellcasting ability to their damage. Notable exceptions are Eldritch Blasts cast by a Warlock with the proper invocation, any evocation spells cast by an Evoker of level 10 and above, some elemental spells cast by a dragon-blooded sorcerer, or some specific spells cast by clerics of some domains.
