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Damage over time is the name given to conditions which do damage over the course of several turns. DOT damage may come in the form of different damage types.  Each type has a respective DoT effect, which go by different names.

Each "Damage over time" condition is given a number (Burn 5, chill 3, Diseased 4, Etc.), which corresponds to a decreasing counter on the affected champion.  At the start of each turn, the affected champion takes damage equal to the number on that counter, and the number then decreases by 1.  This process repeats itself until the counter reaches 0 (Burn 5 becomes Burn 4, then 3, 2, 1).  If the champion is still alive when the DoT counter reaches 1, at the start of the next turn it takes 1 damage and the condition is removed.

Assuming the champion survives, and normal (unreduced) damage takes place, the effect will look like one of the following.

Duration 3:  3+2+1=6 damage over 3 turns. 

Duration 4:  4+3+2+1=10 damage over 4 turns.

Duration 5:  5+4+3+2+1=15 damage over 5 turns.

TriviaAugment

A damage over time effect is frequently referred to as a "DoT" or dot effect.

Some damage over time effects also cause debuffs or additional effects for the duration.

Poison causes -1 DEF and -1 DMG.

Scoured causes -1 DEF.

Diseased causes the condition Inhibited.