Introduced by Eunice Healy, Virginia Bolen, Norton Dean, Kathryn Kimber, Richard Smart, Robert Smith and Tommy Wonder in Broadway revue Two For The Show, the sequel to One For The Money from '39.
Benny Goodman [first recording (Columbia); vocal by Helen Forrest]
Freddy Rich [vocal: Rosemary Calvin]
Barney Bigard [with Shelly Manne]
Les Paul [without Mary Ford on a V-Disc]
Don Byas [influenced Sonny Rollins]
Ella Fitzgerald [and several times afterwards]
Charlie Parker [his Ornitology is also loosely based on How High The Moon]
Les Paul & Mary Ford [n°1 US; first successful guitar and vocals overdubs in recorded history]
Rob De Nijs [as 'n Egoist]
Jeff Beck [vocal: Imelda May]
So many jazz covers the song was nicknamed 'The Bop Hymn'.