Vowel harmony is a type of long-distance assimilatory phonological process involving vowels that occurs in some languages. A vowel or vowels in a word must be members of the same subclass (thus "in harmony"). In languages with vowel harmony, there are constraints on which vowels may be found near each other. Suffixes and prefixes will usually follow vowel harmony rules. Many agglutinative languages have vowel harmony.
The term vowel harmony is used in two different senses.
In the first sense, it refers to any type of long distance assimilatory process of vowels, either progressive or regressive. When used in this sense, the term vowel harmony is synonymous with the term metaphony.
In the second sense, vowel harmony refers only to progressive vowel harmony (beginning-to-end). For regressive harmony, the term umlaut is used. In this sense, metaphony is the general term while vowel harmony and umlaut are both sub-types of metaphony. The term umlaut is also used in a different sense to refer to a type of vowel gradation. This article will use "vowel harmony" for both progressive and regressive harmony.
Harmony processes are "long-distance" in the sense that the assimilation involves sounds that are separated by intervening segments (usually consonant segments). In other words, harmony refers to the assimilation of sounds that are not adjacent to each other. For example, a vowel at the beginning of a word can trigger assimilation in a vowel at the end of a word. The assimilation occurs across the entire word in many languages. This is represented schematically in the following diagram:
In the diagram above, the Va (type-a vowel) causes the following Vb (type-b vowel) to assimilate and become the same type of vowel (and thus they become, metaphorically, "in harmony").
The vowel that causes the vowel assimilation is frequently termed the trigger while the vowels that assimilate (or harmonize) are termed targets. When the vowel triggers lie within the root or stem of a word and the affixes contain the targets, this is called stem-controlled vowel harmony (the opposite situation is called dominant). This is fairly common among languages with vowel harmony and may be seen in the Hungarian dative suffix: