In Western music melismas are often used in operas or any other music which is written for the singer to show off the voice. Melismatic music is the opposite of music where each syllable has its own note. In the "Alleluia Chorus" from Handel's Messiah each syllable of the word "Alleluya" has its own note. But in other parts of the Messiah there are often melismas, e.g. in "Thus saith the Lord" the word "shake" ("and I will shake the heavens and the earth") is set to a run of 25 fast notes. This example can also be described as "word painting". Handel uses the melisma to make it sound like someone shaking something.
Melismas are used a lot in music from many different cultures.