It has been previously stated there are these damage/spell types:

Physical
Fire
Cold
Lightning
Poison
Arcane

With the addition of the Monk class, Holy damage has been introduced. Is Holy damage only going to be applied to the Monk class or may other classes utilize it? And will Holy damage be a type of statistic on weapons, e.g. The Lightning and Cold infused axes we have seen so far.

Additionally are there plans to add Earth or Wind types of damage/spells? The Wizard does have the "Tornado Spell", but what damage type does it fall under? If an element is going to be used as a damage type should it not get the same treatment as other damage types? (See my example on the weapon stats mentioned above.)

Damage types aren't actually completely finalized. There's likely to be some additions, maybe some combinations. For instance people may have noticed that a few of the witch doctor skills refer to a damage type called "black magic". All of the monk spells seen relied purely on weapon damage as their base, but that doesn't mean there couldn't be a holy damage type. What we don't want is for players to carry around a ton of different resistance sets to deal with a large number of different types of damages. There's quite a few different ways to go about solving that, but just to be clear we want damage types to serve the gameplay and not require players to have to pursue complex or item-heavy solutions to it.

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Q u o t e:
On a similar note is there any plans to include damage type on the skills' tooltips?

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Probably. At the very least on skills that it may not be obvious for, which if I were to guess would probably just make it easier to list it for all of them. But, UI/skill description brevity is always a goal, so we'll see.