Weidman was born in New York City, the son of Peggy Wright and librettist and novelist Jerome Weidman. He received a B.A. from Harvard University with a major in East Asian history and a J.D. from Yale Law School.
He has written the books for a wide variety of stage musicals.
He collaborated on three stage musicals with Stephen Sondheim. The first book for a musical that he wrote was Pacific Overtures, as an intern for Hal Prince.The musical premiered on Broadway in 1976, and was revived in 2004 at Studio 54. Assassins first opened Off-Broadway in December 1990 at Playwrights Horizons,  and later opened in the West End in October 1992 at the Donmar Warehouse. The musical was revived on Broadway in 2004 in a Roundabout Theater Company production. Road Show, after several title changes, opened Off-Broadway at the Public Theatre in 2008. The musical had productions at the Goodman Theatre and Kennedy Center in 2003 prior to the Off-Broadway production.
Weidman has worked with choreographer/director Susan Stroman three times. He wrote the book for the musical Big: the musical, with Stroman as choreographer. The musical opened on Broadway in April 1996 at the Shubert Theatre. He and Stroman co-created the Tony Award-winning musical Contact, which opened Off-Broadway in October 1999 at the Newhouse Theatre at Lincoln Center and on Broadway at the Vivian Beaumont Theatre in March 2000. He wrote the book for the new musical Happiness, which ran at Lincoln Center in February 2009. Directed and choreographed by Stroman, the composer was Scott Frankel with lyrics by Michael Korie.
He has been nominated for the Tony Award for Best Book for a Musical three times.
In the 1970s, Weidman wrote for National Lampoon. Since 1986, Weidman has been a writer for Sesame Street, for which he and the writing team have won more than a dozen Daytime Emmy Award for Outstanding Writing For A Children's Series. 
Stage work (libretti)
- Pacific Overtures - 1976 (Tony Award nominee, Best Book of a Musical)
- Anything Goes - 1987 (revised original book with Timothy Crouse; Tony Award, Best Musical Revival)
- Assassins - 1990 (revived on Broadway in 2004) (Drama Desk Award nominee, Outstanding Book of a Musical; Tony Award, Best Musical Revival)
- Big - 1996 (Tony Award nominee, Best Book of a Musical)
- Contact - 1999 - Off-Broadway; 2000 - Broadway (Tony Award nominee, Best Book of a Musical; Tony Award, Best Musical)
- Bounce - 2003
- Take Flight - 2007
- Road Show - 2008
- Happiness - 2009
- "Biography" filmreference.com
- brooklaw listing
- "Storytelling with Sondheim" harvardmagazine.com, January-February 2011
- [Assassins at the Internet off-Broadway Database]
- Michael H. Hutchins (2008). "Assassins". The Stephen Sondheim Reference Guide. Retrieved 2008-08-09.
- Ernio Hernandez. "The Shots Heard Round the World: Musical Assassins Opens on Broadway". Playbill. Retrieved 2015-08-18.
- Contact Broadway ibdb.com, accessed August 19, 2015
- Brantley, Ben. "Accidental Tourists, on a Train to Eternity", The New York Times, March 31, 2009
- Hetrick, Adam. "Arcelus, Foster, Gleason and More Find Happiness at Lincoln Center Starting Feb. 27" Playbill, February 27, 2009
- Internet Movie Database listing
- "PBS, ABC top Emmy nods" Variety, May 12, 2002
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