Curvis was the fourth son of Dai Nancurvis, who had been a bantamweight fighter in the British Army, and had opened a gym in Swansea on leaving the forces.

In 1958, while still an amateur in the Armed Forces, he was selected to represent England in the 1958 British Empire and Commonwealth Games, which that year was held in Cardiff, Wales.

Although it went the full distance, Curvis was knocked down in the sixth, tenth and thirteenth rounds by body punches, and lost a unanimous points decision.

Curvis continued to fight, suffering a defeat against Willie Ludick in Johannesburg In November 1965, he defended his British and Commonwealth titles for the sixth time, against the Scot, Sammy McSpadden in Cardiff, winning by a technical knockout in the twelfth round.

In April 1966, he challenged for the vacant European welterweight title, fighting the Frenchman, Jean Josselin in the Palais des Sports, Paris.

Curvis fought one more fight, defeating Des Rea in Carmarthen in September 1966, before retiring from the ring as undefeated British and Commonwealth welterweight champion.