Played by the gamer-renowned, Video Game Pianist (a.k.a. Martin Leung).
The Entire Super Mario Bros. Soundtrack. (Excluding the Ending Theme.) The track takes a little while to load, because it so, well to be blunt, big.
Thanks to the Video Game Pianist for playing the themes.
The Soundtrack can also be found on Nintendo-Daily. 
The Entire Super Mario Bros. Soundtrack
*** All themes have the prefix "Super Mario Bros."***
- Overworld Theme (1-1)
- Underworld Theme (1-2)
- Castle Theme (1-4)
- Victory Theme (After beating a level)
- Underwater Theme (2-2)
- Star Theme (After acquiring a Starman)
- Ending Theme (The ending. Duh.)