Norwood was born in Alexandria, Virginia and graduated from Thomas Jefferson High School in Alexandria in 1978. He played both football and soccer at James Madison University and graduated with a business degree in 1982.
Norwood quickly became an asset to an offense that was going places as the Bills' general manager, Bill Polian, assembled talent like Jim Kelly, Thurman Thomas, and, on defense Bruce Smith. Within two seasons of Norwood's arrival, the Bills had won the AFC East for the first time since 1980 and made it to the conference championship game. He soon overtook O.J. Simpson as the team's all-time leading scorer. Following the 1990 season, the Bills advanced to their first-ever Super Bowl.
Norwood's field goal range was unusually short for a professional kicker and he had difficulties in converting field goals over 40 yards throughout his career, especially on natural grass (the Bills' home stadium used AstroTurf, which minimized this problem). Super Bowl XXV, which was played on January 27, 1991, would cement Norwood's name in football history when he missed a 47-yard field goal attempt at the end of the game, giving the New York Giants the victory, and started the string of the Bills' four consecutive Super Bowl losses. This kick was made famous by the "wide right" call by the TV announcers.
Although the Bills signed Bjorn Nittmo as Norwood's potential replacement in the 1991 offseason, Norwood remained with the Bills through that season. The Bills returned to the Super Bowl and Norwood was perfect throughout the postseason, including a 44-yard field goal that served as the decisive margin in the AFC Championship Game against Denver.
Norwood was waived in the first roster move of the off-season after the Bills signed Steve Christie, formerly of the Tampa Bay Buccaneers.
After the Bills waived him, Norwood initially returned home to northern Virginia and disappeared completely from the public eye for a number of years, eventually becoming an insurance salesman through the 1990s before returning to Buffalo as a real estate agent in 2002.
Norwood resides with his wife Kimberly in the Washington, DC suburb of Centreville, Virginia. They have three children: twins Carly and Connor (born 1995) and Corey (born 1996).
In the 1994 film Ace Ventura: Pet Detective, a key plot point involves a kicker for the Miami Dolphins named Ray Finkle; in the story, Finkle misses a field goal attempt in the closing moments of Super Bowl XVII, causing the Dolphins to lose the game by a single point - an obvious reference to Norwood's infamous kick in Super Bowl XXV. (In reality, Super Bowl XVII was contested between the Dolphins and the Washington Redskins; the Redskins won 27-17.)
The 1998 film Buffalo '66 features a subplot where main character Billy Brown, played by Vincent Gallo, attempts to murder a former Buffalo kicker named "Scott Wood", whose missed field goal led to Brown losing a large bet and serving a prison term when he took the fall for his bookie.