Part of an ongoing series of 5e (2014) Rules notes.  See the end of the post for notes on 5.5e (2024) rules.

The basic rule: you can only use a spell scroll if you are in a class that has the spell on its spell list.

Things seem pretty simple if you just look at the DMG’s description of Magic Items: Scrolls (DMG 139):

The most prevalent type of scroll is the spell scroll, a spell stored in written form …. A scroll is a consumable magic item. Unleashing the magic in a scroll requires the user to read the scroll. When its magic has been invoked, the scroll can’t be used again. Its words fade, or it crumbles into dust.

Unless the scroll’s description says otherwise, any creature that can understand a written language can read the arcane script on a scroll and attempt to activate it.

However, under Spell Scroll (DMG 200), the process is much more elaborate and restrictive (and in D&D, specific beats general):

A spell scroll bears the words of a single spell, written in a mystical cipher.

If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components. Otherwise, the scroll is unintelligible.

Casting the spell by reading the scroll requires the spell’s normal casting time. Once the spell is cast, the words on the scroll fade, and it crumbles to dust. If the casting is interrupted, the scroll is not lost.

If the spell is on your class’s spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + the spell’s level. On a failed check, the spell disappears from the scroll with no other effect.

That page also includes a table for determining the saving throw DC and attack bonus:

Spell Level	Rarity	Save DC	Attack Bonus
Cantrip	Common	13	+5
1st	Common	13	+5
2nd	Uncommon	13	+5
3rd	Uncommon	15	+7
4th	Rare	15	+7
5th	Rare	17	+9
6th	Very rare	17	+9
7th	Very rare	18	+10
8th	Very rare	18	+10
9th	Legendary	19	+11

Spell scrolls can also serve as fodder for a spell book.

A wizard spell on a spell scroll can be copied just as spells in spellbooks can be copied. When a spell is copied from a spell scroll, the copier must succeed on an Intelligence (Arcana) check with a DC equal to 10 + the spell’s level. If the check succeeds, the spell is successfully copied. Whether the check succeeds or fails, the spell scroll is destroyed.

If you want to read a discussion of whether Spell Scrolls need to actually be scrolls, check here.

What if I just want to know what is on the scroll, just not cast it (yet)?

Since we don’t have Read Magic any more in D&D, how do we know what is on a scroll? That’s actually … a question without a very clear answer.

The Identify spell will do it. But short of that, the answer is, “It depends.”

If the spell scroll is just a recipe for the spell, then the normal rules of (1) reading scrolls and (2) identifying what it does apply:

• you need to be able to read
• you need to be able to cast the spell in order to read it (i.e., it has to be in your class spell list)
• and you need to spend a Short Rest trying to puzzle it out, just like any other magic item.

A beneficent, organized, communicative spellcaster, in forming the scroll, might have put a label on it (“Spell of Fireball” in plaintext). In which case you’d have a pretty big clue as to what it is and does, assuming you could find a beneficent, organized, communicative spellcaster’s works. And that you could actually trust that was what it does.

In theory, you could just cast the spell by reading it for the first time, without actually knowing what it does until the very end. A charitable GM might even let you make some sort of roll (e.g., Intelligence (Arcana) vs 10 + spell level) if, as you realize at the last moment what it does, you wanted to abort casting it. (It would still suck up a turn’s Action, though, as a minimum cost.)

I would also be willing to entertain the idea that, if you simply spend a Short Rest focusing on a scroll, you should be able to get an impression of what it does even if you could not use it and/or read it. A sense of the type of magic (necromantic, evocation), aspects of it (heat, cold, water, steel), colors, a usable class (choirs singing, the smell of damp earth), that sort of thing.

Or maybe not. Since you cannot actually read the scroll without being able to cast it, it sort of plays like “language” (“Crap, this thing is in German. Anyone know German?”) … but it’s definitely not a language. I mean, it’s possible to have a scroll that is usable (intelligible) to a druid and a sorcerer,  and second one to a sorcerer and a wizard, and a third to a wizard and a druid, and language simply doesn’t work like that. Instead, it’s as though the words and formulae tie into some sort of internal mindframe, some perception of reality, that is shared within some magic-using classes in some ways, but not non-magic-using classes (except, sorta, Rogues).

So more like, “Crap, this one is giving me a migraine looking at it, someone else want to give it a go?” Which might be the quickest way to deal with spell scrolls when found during an adventure, just having the various magic-users in the party pass each of them around until someone can read it. That is, you can quickly (if maybe painfully) tell if your class can use the spell, though you’ll need to spend that Short Rest to determine what precisely it is.

(Some interesting discussion here about this whole sub-question.)

What about Thieves?

Thieves are (in some cases) a weird exception to the above. At 13th level, Thief Rogues get “Use Magic Device” ability (PHB 97), giving them access to magical devices they would not be able to otherwise access.

By 13th level, you have learned enough about the workings of magic that you can improvise the use of items even when they are not intended for you. You ignore all class, race, and level requirements on the use of Magic Items.

This includes spell scrolls, per the Sage Advice Compendium:

Does the Thief’s Use Magic Device feature allow them to use spell scrolls? Yes. The intent is that a Thief can use spell scrolls with Use Magic Device

The thief would still have to make the ability check to actually cast the spell successfully, with the spellcasting ability = 0 (vs a DC of 10 + spell level), and without any proficiency bonus added in (basically a straight d20). If the spell requires a further spell attack roll, again the spellcasting ability is 0, but proficiency bonus does apply.

Do I have to Concentrate if I use a Spell Scroll to cast a spell that requires Concentration?

Yes. As the basic rules say (emphasis mine):

Some magic items [such as spell scrolls] allow the user to cast a spell from the item. The spell is cast at the lowest possible spell level, doesn’t expend any of the user’s spell slots, and requires no components, unless the item’s description says otherwise. The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires Concentration. Many items, such as potions, bypass the casting of a spell and confer the spell’s effects, with their usual duration. Certain items make exceptions to these rules, changing the casting time, duration, or other parts of a spell.

So scrolls give you the advantage of no components and no spell slots required. But you still have to concentrate/control the spells they cast.

Does any of this change in 5.5e?

The 5.5e (2024) edition rules do change some of the above.

Scrolls are now cast using the Magic Action in combat (which lets you cast a spell, or use a feature or magic item that requires a Magic action to activate).

The DMG (216) defines magical scrolls (beyond their physical description) as:

The most prevalent scroll is the Spell Scroll, a spell stored in written form. However, some scrolls, like the Scroll of Protection, bear an incantation that isn’t a spell. 

Great.

Just like in 5e, when reading the scroll to unleash its power, the scroll itself (or the writing on it) is destroyed.  The section concludes (on DMG 217; emphasis mine):

Any creature that can understand a written language can read a scroll and attempt to activate it unless its description says otherwise.

That sounds like the wide-open-reading clause in the 5e DMG.  However, if you look up Spell Scroll in the 5.5e PHB under (ch. 6) Equipment > Adventuring Gear, you find …

If the spell is on your class’s spell list, you can read the scroll and cast the scroll using its normal casting time and without providing any Material components. If the spell requires a saving throw or attack roll, the spell save DC is 13, and the attack bonus is +5.

There’s nothing about spell scrolls in the Magic Items section of the PHB (p. 233), but in the Crafting rules right afterwards, it does talk about creating spell scrolls, much more clearly than 5e did. 

• A table is provided showing the time and cost of doing so (based on the spell level).
• The scribe has to have Proficiency in Intelligence (Arcana) or in Calligrapher’s Tools, and must prep the spell each day while they are writing the scroll. 
• Any material items are consumed when the scroll is completed.
• For leveled spells, the Scroll’s spell uses Spell Save DC and Spell Attack Bonus of the scribe. For cantrips, the Scroll’s spell works as though the caster were the scribe’s level.

Spell Scrolls also come up in the Wizard class description [PHB 167], talking about expanding a Wizard’s Spellbook:

You could discover a Wizard spell on a Spell Scroll, for example, and then copy it into your spellbook. […]  When you find a level 1+ Wizard spell, you can copy it into your spellbook if it’s of a level you can prepare and if you have time to copy it. For each level of the spell, the transcription takes 2 hours and costs 50 GP. Afterward you can prepare the spell like the other spells in your spellbook.

Unlike in 5e, so far as I can tell, no Arcana roll is needed, nor is the Spell Scroll destroyed by doing so.

Thief and Use Magic Device

For this completely reworked Level 13 subclass feature, it notes (among other things):

You can use any Spell Scroll, using Intelligence as your spellcasting ability for the spell. If the spell is a cantrip or Level 1 spell, you can cast it reliably. If the scroll contains a higher-level spell, you must first succeed on an Intelligence (Arcana) check (DC 10 plus the spell’s level). On a successful check, you cast the spell from the scroll. On a failed check, the scroll disintegrates.