The 58-year-old, who has been out of work since leaving Southend United in January, has been given the job on a short-term contract.
Swindon have been without a permanent manager since David Flitcroft left the role on March 1 to take over at Mansfield Town.
After putting pen to paper at the County Ground, Brown told the club's official website: "It's a great challenge. It's a short-term deal at the moment and the lure of the playoffs and possible promotion was a big attraction for me, as was the sales pitch from the chairman. Lee [Power] outlined what he wants for the club and the ambition and that really interested me."
Brown will be joined by Matthew Taylor in his coaching staff.