Felix Mendelssohn's "Wedding March" is one of the best known of the pieces from his suite of incidental music (Op. 61) to Shakespeare's play A Midsummer Night's Dream, written in 1842

The famous Wedding March is not actually played in A Midsummer Night's Dream. It is commonly associated with weddings due to its use in Felix Mendelssohn's incidental music for the play.

Mendelssohn: Midsummer Night's Dream

Richard Wagner, the music is from "A Midsummer Night's Dream".

The usual recessional used at weddings is from "Lohengrin" and was written by Felix Mendelssohn.

It's the other way round. Wagner wrote the music known as 'Here comes the bride' and Mendelssohn's Wedding March is from his incidental music to A Midsummer Night's Dream

Not the way Shakespeare wrote it, but Felix Mendelssohn wrote some absolutely terrific incidental music to accompany the play, including an overture and a wedding march which I guarantee you have heard before. Mendelssohn's music features prominently in both the 1935 and 1999 movies and the overture can be heard at the related link.