Fleck started his senior career with Rangers under manager Jock Wallace, although it was only after the arrival of Graeme Souness as player/manager in 1986 that he began to make a real impact, scoring 22 goals in 48 appearances in season 1986–87. Fleck's goal tally that season included four hat-tricks; in league games against Clydebank and Falkirk, and in a UEFA Cup tie against Ilves Tampere. In December 1987 Fleck was transferred for £580,000 to Norwich City in the EnglishFootball League First Division. He scored 66 goals in 181 appearances in his first spell with the club, earning a call-up to the Scotland Squad, winning 4 caps, including 2 at the 1990 World Cup. 'Flecky' attained hero status among Norwich supporters and was voted Norwich City player of the year in 1992 which was the final season of his first spell at Carrow Road. He helped them finished fourth in the league in 1989, when they also reached the FA Cup semi-final, though Fleck and his team-mates were unable to compete in the following season's UEFA Cup due to the ongoing ban on English clubs in European competitions that followed the Heysel disaster of 1985. Fleck helped the Canaries reach another FA Cup semi-final in 1992, where they surprisingly lost to Second Division underdogs Sunderland, Fleck returned from injury to play in this game but he was clearly unfit. He controversially moved on to Chelsea for a club record fee of £2.1million just before the first season of the new Premier League got underway, Norwich turned down the bid, but he forced it through by threatening to go on strike and refuse to play for the club again. At the time the deal was a record sale for Norwich, As part of the deal a clause was included to ruled Fleck out of playing against Norwich the 1st time they met just a couple of weeks later. Despite his now legendary status at Norwich, at the time he was probably one of the most hated players among Norwich fans. The move proved unsuccessful; Fleck scored just four goals in 48 appearances and was loaned out to Bolton Wanderers and Bristol City. He missed out on a place in Chelsea's squad for the 1994 FA Cup Final, which they lost 4–0 to Manchester United. Despite his dismal goalscoring record at Chelsea, he is remembered fondly by the club's fans, who sang a song in his honour – We all live in a Robert Fleck world – based on the lyrics of Yellow Submarine by The Beatles. Fleck re-joined Norwich for £650,000 in September 1995 after a loan spell, to begin with he made an impact forming a solid partnership with Ashley Ward up front, but after Ward was sold due to Norwich's financial problems, his performance was no where as consistent as in his first spell at Carrow Road, and they failed to make a serious attempt to push for promotion in any of his three seasons back at the club. He moved to Reading towards the end of the 1997–98 season, spending one season at the Berkshire club, and in nine league games he scored once against Luton Town, before injury ended his playing career.
Managerial career
After a spell as player-manager with Gorleston, where he won the Norfolk Senior Cup in 2001, he joined Diss Town who play in the Ridgeons League, as manager in the summer of 2002. He won the Norfolk Senior Cup again in 2003 and 2005. A poor run of form during 2006 saw him sacked on 18 October 2006. Fleck returned to employment with Norwich City in May 2007, when he joined the club's scouting network.
Teaching career
Fleck is now a teaching assistant at the Parkside School, "a Norwich school for children with complex needs", a career move that developed from running coaching sessions for children at the school.
Fleck is married to a "Norfolk girl" and has an adult daughter who is a beautician. While playing, Fleck funded trips for 18 months for a child with a terminal condition. Fleck's nephew John plays for Sheffield United.