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Defense
The Normal type is a mixed bag defensively. It sports only one weakness, but only one immunity. A fair number of Normal-type Pokémon also have high stats defensively. However, Fighting-type moves are very common and have gained popularity due to their advantage over the Normal type. This is made even worse by the fact that Generation V has introduced many common powerful Fighting-type Pokémon, as well as more powerful moves.
Offense
The type performs similarly offensively. There are only two types that resist Normal-type moves (Rock and Steel), and one type that is immune (Ghost). Normal-type moves cannot deal super effective damage at all though. This is somewhat countered by many Normal types being able to learn Fighting and Ground moves, which are super effective against both types that are resistant to Normal-type moves (excluding Ghost).