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I had a player try a mutiny tonight. We were running a battle against a bunch of Red Wizards of Thay, and I chose to cast fireballs by the boat load. We are all new to D&D and one proceeded to tell me that fireball and other AOE spells require a "to hit" roll like magic missile or fire bolt.

I stated that since the spell description doesn't ask for a ranged attack roll, it hit automatically and only their Dex save would help them.

Am I right in stating that AOE spells, unless otherwise stated in the spell description, do not require a roll against the target's AC?

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No, it does not.

To resolve a spell, you just read the spell's description and follow those instructions. Unless it explicitly asks for an attack roll, you don't make one. See the chapter on Spells (page 205 in the PHB, or here in the basic rules):

Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target.

Emphasis mine.

Fire bolt, for example, reads:

Make a ranged spell attack against the target.

But fireball says:

Each creature in a 20-foot-radius sphere centered on that point must make a Dexterity saving throw.

It mentions nothing about an attack roll, so you don't need to make one.

There also exist some spells that require both, such as ice knife (XGtE, p. 157; EEPC, p. 19):

Make a ranged spell attack against the target. On a hit, the target takes 1d10 piercing damage. Hit or miss, the shard then explodes. The target and each creature within 5 feet of it must succeed on a Dexterity saving throw...

But such spells will explicitly say so; fireball does not.

Note that magic missile says nothing about either of these things, so you don't have to do either. It just hits automatically.

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