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Evolution is when Pokémon change form. There are many different ways this happens:

Leveling-up

Sometimes when a Pokémon levels up it evolves. To grow levels, your Pokémon must get a certain amount of experience points (EXP). Obviously the only way to do that is to have it participate in battle (EXP share also works).

Trading

Using a GBA link cable you can trade certain Pokémon to other trainers. Some Pokémon evolve when this happens.

Items

When some Pokémon are given items they evolve.

Trading with items

Rare items are known to evolve Pokémon, but only if they are traded with the item.

Stones

Leaf, Sun, Moon, Fire, Thunder and Water Stones also evolve Pokémon, but these are rare.

Random

Random evolution occurs when one Pokémon may turn into one of two different Pokémon.

Specific conditions

When it doesn't fit one of the other categories, it is considered a special condition.

Taming

When Pokémon are happy enough, they might evolve. In Verdanturf Town there is a lady who can rate your Pokémon's happiness.

Spectral

Once one Pokémon becomes two, it enters this category.

Combination

If it fits two or more categories it goes here.