The Newmarket-based handler Yorkshireman recorded a famous success at Doncaster in 1993, when Bob's Return won the St Leger in the hands of Philip Robinson.
Bob's Return was also victorious in the Great Voltigeur Stakes and was second in the Eclipse and Hardwicke Stakes.
Tompkins recorded many other big wins, and went close to landing the 2000 Guineas when Even Top was beaten a short head by Mark Of Esteem at Newmarket in 1996.
Oscar-winning actress Dame Judi Dench enjoyed some great days with the Tompkins yard, owning Smokey Oakey in partnership with her driver Bryan Agar and seeing him win the Brigadier Gerard Stakes and Lincoln Handicap.
Ted Spread was another popular Tompkins-trained runner, winning the Chester Vase in 2010.
Tompkins said: "I'm 68 and have decided to retire, I've not been very well and it's getting harder and harder for everybody.
"I've been training since 1979 and trying my best - we've had good horses and good owners, but it gets harder to find the good horses. They are with a small pool of trainers.
"A bit like Luca (Cumani, who retired in 2018), I've got my stud and will live there."
Looking back on his career, Tompkins said: "Bob's Return only cost 14,500 (Irish pounds) and was a proper horse. The day he won the Leger was a fantastic day and he won well.
"Halkopous won the Fighting Fifth and the Bula, he was a cracking horse, Smokey Oakey, Babodana, Cool Edge, who only cost 750 pounds in Irish money.
"It's been great, we've done it for 40 years and had a fantastic time."
He added: "We've one or two to run, but I'll certainly finish at the end of the month. A few will go to James Eustace, my former assistant, my owners know him so hopefully a few will go there."
