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Basically, if you are a wizard with the war caster feat carrying a shield and you cast shield, do you get the bonus of your physical shield and the bonus from the spell shield (+5 AC) or not?

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All bonuses to AC stack.

I assume you're thinking of previous versions of D&D, where bonuses of the same type didn't stack. However, in 5e, the majority of ways to gain AC are separate ways to calculate your base AC, such as Plate Armour, which sets your base AC to 18, or Mage Armour, which sets your base AC to 13 + Dex. From the Player's Basic Rules, page 9:

Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.

Actual bonuses to AC are very rare in 5e, and stack fully with each other and with different sources of AC, unless the source or the bonus specifies otherwise. An example of this is the Monk's Unarmoured Defense ability, which specifies that it only applies when you are not carrying a shield. Note that it says nothing about the Shield spell, so that would function normally.

An example of non-stacking bonuses is the physical shield itself, which specifies that you can benefit from only one shield at a time. So carrying 2 shields only gets you a +2 bonus. (Player's Basic Rules, pg 44.) Note that the Shield spell provides you with 'an invisible barrier of magical force', not a (lowercase) shield. (Player's Basic Rules, pg 100.) So it does not come under this exception and stacks with a physical shield without difficulty.

So a shield stacks with the Shield spell, and both of them stack with any armour you are wearing, or Mage Armour if you have that instead.

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