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I'm a new DM and I have a relatively simple question that I can't seem to find a good answer to: How do I determine how many spells a character gets to learn upon leveling up?

For example: The table for a wizard shows that a level one wizard knows 3 cantrips and gets 2 level one spell slots. Does this also mean they know 2 level one spells? Now, suppose they level up to level 2. They get an extra level one spell slot, so do they get an extra level one spell? At level 3, it gets more confusing, because they now get an extra level one spell slot and 2 new level 2 slots. How many spells does the wizard now know?

More generally, is there a way to figure out how many spells a character is able to learn per level which applies to all classes? Does it depend on class?

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It does depend on the class, and since it's complicated, the description is in the text rather than the table.

  1. Wizards don't really know spells (except for cantrips) — they add them to their spellbooks, which can happen while adventuring in addition to those they learn on level-headed. This is under "Learning Spells of 1st level or higher" in the book.
  2. Most "divine" casters — clerics, paladins, druids — don't learn individual spells . They know all of them (but prepare a limited subset per day).
  3. Other casters — sorcerers, rangers, warlocks, bards — do have a fixed number of Spells Known, and in this case, you will find it on the class table. See for example the sorcerer or ranger.
  4. This is also true for the martial and roguish archetypes which add spellcasting — each of these have a special second table. For example, Eldritch Knight (link will work if you own the PHB on D&D Beyond; otherwise, check your paper copy).

(This isn't a duplicate, but answers turn out to be similar because the same thing needs to be explained: What happens when I can prepare more spells than I have slots to cast them with?)

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