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So!

I didn't find anything in the D&D 5e books about learning spells as an arcane spellcaster.

I have a question:

  • As, say, a Wizard levels up and gets new spell slots, does he need to go to a library and find the spell somewhere? Could he learn a spell he saw someone else cast?
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    \$\begingroup\$ Related: How does one find new spells? \$\endgroup\$ – Adeptus Sep 18 '15 at 2:36
  • \$\begingroup\$ I put it that way because there's other arcane spellcasters in the game. I understand that a divine caster such as a cleric would receive spells from the deity, and a ranger or druid would get them through their connection with nature. A sorcerer has innate casting ability so I'd think of them as D&D Jedi... but a Bard? Same as a wizard, I'd guess, since he's also an arcane caster. \$\endgroup\$ – Ulrich Sep 18 '15 at 2:48
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The Wizard

At leveling up you are allowed to simply choose two spells from the wizard list and declare that they are now in your spellbook. (PHB p.114: Learning Spells of 1st Level and Higher)

But there's more! You can build up your spellbook much quicker than that: see the sidebar "Your Spellbook" on p. 114 of the PHB. This describes the sort of process you mention, whereby you add to your spellbook by studying another wizards' copy, reading a scroll, or even through original research. (This, in my opinion, is the sine qua non of the wizard: the ability to build up such a large repertoire from which to prepare every day.)

See also this answer to Is it possible for a wizard to change the spells in her spellbook?

The Bard

@DaleM's excellent answer:

You know four 1st-level spells of your choice from the bard spell list.

The Spells Known column of the Bard table shows when you learn more bard spells of your choice. Each of these spells must be of a level for which you have spell slots, as shown on the table. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.

Additionally, when you gain a level in this class, you can choose one of the bard spells you know and replace it with another spell from the bard spell list, which also must be of a level for which you have spell slots.

But there's no more! Unlike the wizard, you can never know more spells than this.

The wizard is unique among spell casters in being able to acquire more spells by finding them. Some casters like Clerics have access to all of the spells on the list when preparing them and others like Bards have a limited and immutable list of spells that they know.

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@nitsua60 has covered the wizard; for the bard (PHB p.53):

You know four 1st-level spells of your choice from the bard spell list.

The Spells Known column of the Bard table shows when you learn more bard spells of your choice. Each of these spells must be of a level for which you have spell slots, as shown on the table. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.

Additionally, when you gain a level in this class, you can choose one of the bard spells you know and replace it with another spell from the bard spell list, which also must be of a level for which you have spell slots.

But there's no more! Unlike the wizard, you can never know more spells than this.

The wizard is unique among spell casters in being able to acquire more spells by finding them. Some casters like Clerics have access to all of the spells on the list when preparing them and others like Bards have a limited and immutable (except for the one spell you can change each level) list of spells that they know.

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    \$\begingroup\$ It feels kind of weird to upvote two incomplete answers because each references and relies on the other. Isn't the standard to incorporate the other answer, so that each answer stands alone? \$\endgroup\$ – GMJoe Sep 18 '15 at 5:09
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    \$\begingroup\$ @GMJoe no, we can't allow either answer to become too powerful. \$\endgroup\$ – Liesmith Sep 18 '15 at 10:13
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    \$\begingroup\$ @GMJoe I'd first answered the question when there was no reference to bards. I'd started to edit my answer to include bards when Dale M posted his perfectly good bard-answer. I saw no need to steal Dale's thunder, as it were, and just referenced his. Just a weird artifact of timing and a shifting question.... \$\endgroup\$ – nitsua60 Sep 18 '15 at 10:48
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    \$\begingroup\$ So that this is a real answer on its own, it's preferable that it answer the question independently by including any necessary statements, and refer to other answers only to give due credit. \$\endgroup\$ – SevenSidedDie Sep 18 '15 at 17:13
  • \$\begingroup\$ Just a note but the Bard's spells known are not immutable. When they learn a new spell, they can also forget an old spell and replace it with a new one from their spell list that they have spell slots for. \$\endgroup\$ – user27327 Apr 13 '17 at 13:34

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