Damage (Japanese: ダメージ damage) is a loss of a Pokémon's HP that happens as the result of a physical or special attack used against it by another Pokémon.
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Damage calculation
Except for moves that deal direct damage, the damage dealt when a Pokémon uses a damaging move depends on its level, its effective Attack or Special Attack stat, the opponent's effective Defense or Special Defense stat, and the move's effective power. In addition, various factors of damage modification may also affect the damage dealt.
More precisely, damage is calculated as
where
 Level is the level of the attacking Pokémon (or twice the level for a critical hit in Generation I).
 A is the effective Attack stat of the attacking Pokémon if the used move is a physical move, or the effective Special Attack stat of the attacking Pokémon if the used move is a special move (ignoring all^{Gen. II}/negative^{Gen. III+} stat stages for a critical hit).
 D is the effective Defense stat of the target if the used move is a physical move or a special move that uses the target's Defense stat, or the effective Special Defense of the target if the used move is an other special move (ignoring all^{Gen. II}/positive^{Gen. III+} stat stages for a critical hit).
 Power is the effective power of the used move.
and Modifier is
where
 Targets is 0.75 if the move has more than one target, and 1 otherwise. (In Generation III, it is 0.5 for moves that target all adjacent foes with more than one target, and 1 otherwise.)
 Weather is 1.5 if a Watertype move is being used during rain or a Firetype move during harsh sunlight, and 0.5 if a Watertype move is used during harsh sunlight or a Firetype move during rain, and 1 otherwise.
 Badge is applied in Generation II only. It is 1.25 if the attacking Pokémon is controlled by the player and if the player has obtained the Badge corresponding to the used move's type, and 1 otherwise.
 Critical is applied starting in Generation II. It is 2 for a critical hit in Generations IIV, 1.5 for a critical hit from Generation VI onward, and 1 otherwise.
 random is a random factor between 0.85 and 1.00 (inclusive):
 From Generation III onward, it is a random integer percentage between 0.85 and 1.00 (inclusive)
 In Generations I and II, it is realized as a multiplication by a random uniformly distributed integer between 217 and 255 (inclusive), followed by an integer division by 255
 STAB is the sametype attack bonus. This is equal to 1.5 if the move's type matches any of the user's types, 2 if the user of the move additionally has Adaptability, and 1 if otherwise.
 Type is the type effectiveness. This can be 0 (ineffective); 0.25, 0.5 (not very effective); 1 (normally effective); 2, or 4 (super effective), depending on both the move's and target's types.
 Burn is 0.5 (from Generation III onward) if the attacker is burned, its Ability is not Guts, and the used move is a physical move (other than Facade from Generation VI onward), and 1 otherwise.
 other is 1 in most cases, and a different multiplier when specific interactions of moves, Abilities, or items take effect:
Move Value Detail Astonish 2 In Generation III only, if this is the used move, and the target had previously used Minimize Aurora Veil 0.5 * If in effect on the target's side, the move is not a critical hit, and the user's Ability is not Infiltrator (does not stack with Light Screen and Reflect) Body Slam 2 From Generation VI onward, if this is the used move, and the target had previously used Minimize Dragon Rush 2 From Generation VI onward, if this is the used move, and the target had previously used Minimize Earthquake 2 From Generation V onward, if this is the used move, and the target is in the semiinvulnerable turn of Dig* Extrasensory 2 In Generation III only, if this is the used move, and the target had previously used Minimize Flying Press 2 If this is the used move, and the target had previously used Minimize Heat Crash 2 From Generation VI onward, if this is the used move, and the target had previously used Minimize Heavy Slam 2 In Generation VII, if this is the used move, and the target had previously used Minimize Light Screen 0.5 * From Generation III onward, if in effect on the target's side, the move used is special, not a critical hit, and the user's Ability is not Infiltrator Magnitude 2 From Generation V onward, if this is the used move, and the target is in the semiinvulnerable turn of Dig* Needle Arm 2 In Generation III only, if this is the used move, and the target had previously used Minimize Phantom Force 2 If this is the used move, and the target had previously used Minimize Reflect 0.5 * From Generation III onward, if in effect on the target's side, the move used is physical, not a critical hit, and the user's Ability is not Infiltrator Shadow Force 2 If this is the used move, and the target had previously used Minimize Stomp 2 In Generation III and from Generation V onward, if this is the used move, and the target had previously used Minimize* Surf 2 If this is the used move, and the target is in the semiinvulnerable turn of Dive* Whirlpool 2 If this is the used move, and the target is in the semiinvulnerable turn of Dive* ZMoves 0.25 If the used move is a damaging ZMove the target has protected against
Ability Value Detail Fluffy 0.5 If the target has this Ability, and the used move makes contact and is not Firetype 2 If the target has this Ability, and the used move is Firetype and does not make contact Filter 0.75 If the target has this Ability and the used move is super effective (Type > 1) Friend Guard 0.75 If an ally of the target has this Ability Multiscale 0.5 If the target has this Ability and is at full health Prism Armor 0.75 If the target has this Ability and the used move is super effective (Type > 1) Shadow Shield 0.5 If the target has this Ability and is at full health Sniper 1.5 If the attacker has this Ability and the move lands a critical hit Solid Rock 0.75 If the target has this Ability and the used move is super effective (Type > 1) Tinted Lens 2 If the attacker has this Ability and the used move is not very effective (Type < 1)
Item Value Detail Chilan Berry 0.5 If held by the target, and the used move is Normaltype Expert Belt ~1.2 If held by the attacker and the move is super effective (Type > 1) Life Orb ~1.3 If held by the attacker Metronome >1 1 + (~0.2 per successful consecutive use of the same move) if held by the attacker, but no more than 2 Typeresist Berries 0.5 If held by the target, the move is of the resisted type and super effective (Type > 1)
 If multiple effects influence the other value, their values stack multiplicatively. For example, if both Multiscale and a Chilan Berry take effect, other is
0.5 * 0.5 = 0.25
.
During the calculation, any operations are carried out on integers internally, such that effectively each division is a truncated integer division (rounding towards zero, cutting off any decimals), and any decimals are cut off after each multiplication operation. If the calculation yields 0
, the move will deal 1 HP damage instead; however, in Generations I and V, different behavior may occur due to apparent oversights:
 In Generation I, if the calculation yields
0
because the target has two types that both resist the move's type, the move will miss as if it is ineffective;  In Generation V, a move may deal 0 HP damage when other is less than
1
, because the routine to prevent 0 HP damage is erroneously performed before applying the other factor.
Example
Imagine a level 75 Glaceon that does not suffer a burn and holds no item with an effective Attack stat of 123 uses Ice Fang (an Icetype physical move with a power of 65) against a Garchomp with an effective Defense stat of 163 in Generation VI, and does not land a critical hit. Then, the move will receive STAB, because Glaceon's Ice type matches the move's: STAB = 1.5. Additionally, Garchomp is Dragon/Ground, and therefore has a double weakness to the move's Ice type: Type = 4. All other (nonrandom) factors of Modifier will be 1. This effectively gives
When also plugging in Glaceon's effective Attack stat and Garchomp's effective Defense stat, the whole damage formula gives
That means Ice Fang will do between 168 and 196 HP damage, depending on luck.
If the same Glaceon holds a Muscle Band and its Ice Fang lands a critical hit against Garchomp, it will be Critical = 1.5, effectively giving
Furthermore, Ice Fang's effective power will be boosted by the Muscle Band by (approximately) 10% to become 71, leading the whole formula to give
That means Ice Fang will now do between 268 and 324 HP damage, depending on luck.
Pokémon GO
In Pokémon GO, damage is calculated differently due to different variables existing in the game.
where
 Power is the power of the move used
 Attack is the Attack stat of the attacking Pokémon
 Defense is the Defense stat of the Pokémon being attacked
and Modifier is
where
 Type is the type effectiveness, which is calculated differently in GO, using multipliers of base 1.6 instead of 2.
 STAB is the sametype attack bonus. This is equal to 1.2 if the move's type matches any of the user's types, and 1 if otherwise.
 The following variables are applied in Gym and Raid Battles only, and are 1 otherwise.
 Weather is 1.2 if the move used has a weatherboosted type, and 1 otherwise.
 Friendship is applied when battling with Friends and varies depending on the Friendship level.
 1.03 if Good Friends
 1.05 if Great Friends
 1.07 if Ultra Friends
 1.1 if Best Friends
 1 if otherwise
 Dodged is 0.25 if the attack was successfully dodged, and 1 if otherwise.
 Only a Gym defender or Raid Boss's attack may be dodged.
 The following variables are applied in Trainer Battles only, and are 1 otherwise.
 Trainer is 1.3 for all attacks used in a Trainer Battle.
 Charge is applied for Charged Attacks, and its value depends on the player's score during the minigame
 1 if "Excellent!"
 0.75 if "Great!"
 0.5 if "Nice!"
 0.25 if otherwise
By the damage formula, moves that have a power of 0 such as Splash and Yawn will still do 1 HP damage.
Trivia
 In Pokémon Emerald, if the player's Pokémon deals over 33037 HP damage, the HP bar won't be drained. However, if it deals exactly 33037 HP, the HP bar will be drained automatically.
 In Generations V through VII, the amount of damage that can be dealt in a single attack is capped at 65535. In addition, during the calculation of very high damage amounts an overflow can occur during the calculation, causing the actual damage dealt to be much lower than expected.^{[1]}
 In Pokémon Battle Revolution, the HP bar will change with a different animation depending on the move's type (recovery, recoil damage and indirect damage use the Normaltype animation), as shown below.

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References
This game mechanic article is part of Project Games, a Bulbapedia project that aims to write comprehensive articles on the Pokémon games. 