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I'm a pretty new player, and I'm building a bard in D&D 5e. I understand how saving throws work in general, but I am a little confused about one thing. Say I cast Blindness/Deafness on a humanoid. They must then make a Constitution saving throw. My question is: what value do they have to roll in order to make the save? Does their roll (plus their constitution modifier) have to beat my constitution (plus my modifier)? This is more of a general player question than one specific to the Bard or that spell.

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Saving throws are always made against a DC, and in the case of casting spells there is a formula to work out the required DC. From the Basic Rules on spellcasting:

Saving Throws

Many spells specify that a target can make a saving throw to avoid some or all of a spell's effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure.

The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.

And as the Bard class description says:

Spellcasting Ability

Charisma is your spellcasting ability for your bard spells. Your magic comes from the heart and soul you pour into the performance of your music or oration. You use your Charisma whenever a spell refers to your spellcasting ability. In addition, you use your Charisma modifier when setting the saving throw DC for a bard spell you cast and when making an attack roll with one.

Spell save DC = 8 + your proficiency bonus + your Charisma modifier

So your bard spells have save DC based on your charisma modifier and proficiency bonus as described above. In general, whenever a class or racial feature allows you to cast spells somehow, its description will clarify what the relevant spellcasting ability modifier is, if any, so that you can work out the saving throw DC.

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If the roll of the dice + appropriate modifiers meets the DC, it is counted as a success. I always remember, "meets it, beats it". :)

So in terms of this, your spell save DC is what they have to beat. That is calculated at 8 + proficiency bonus + spellcasting mod (charisma for a bard). The highest this can be is 19 (8 + 6[prof] + 5[cha]).

The save is a constitution save, so your target must roll a d20 and add their con modifier to that. If their roll + modifier is the same or higher than your spell save DC, then the spell has been surpassed and the effects do not work.

For some spells, like fireball, the save is only for half damage. So if the target fails the save, they would take the full damage, but on a successful save (they beat your spell save DC) they would take half damage. In terms of blindness/deafness, your spell would have no effect if they succeeded your save.

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