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Mesa was born Jesus Fernández. Prior to beginning his career, and for undisclosed reasons, he changed his name officially to Juan Meza.
He lost for the first time on his second bout, being beaten on points after four rounds by Valentin Holguin on October 13 in Los Angeles. Six days later, he knocked out Davey White in Las Vegas. Eight days later, he had a rematch with Holguin, over the extremely rare five rounds schedule distance. He avenged his first loss by outpointing Holguin.
On November 3, he scored his first knockout in the first round, defeating Horacio Pintado in Los Angeles. But, in his next fight, on November 18, he lost by a four-round decision to Carlos Ortiz, in San Diego.
Meza began rising in boxing's ranks when he knocked Ortiz out in the first round on December 16, also at San Diego. He won 29 of his next 31 bouts, earning fame, particularly in the West Coast of the United States, as aforementioned. Among those thirty one bouts, there were many highlights, including:
- His first presentation as a professional in Mexico, on February 12, 1978, when he knocked out Julio Gaxiola in his hometown of Mexicali.
- A ten-round decision loss to William Berry on August 2, 1978. in Las Vegas.
- A ten-round decision win against experienced James Martinez, who came into their fight sporting a record of 41 wins, 7 losses and one draw (tie), on January 9, 1979, in Phoenix.
- A ten-round split decision loss to Carlos Mendoza, who also fought Roberto Durán and Wilfredo Gómez; among others, on June 26, 1980 in Los Angeles.
- A ninth-round knockout win over Javier Flores, who had 29 wins and 9 losses before their fight, on March 26, 1980, in Las Vegas.
- A ten-round decision win in a rematch with James Martinez, on May 28, 1981 in Las Vegas.
- A ten-round decision win over Carlos Mendoza, thus avenging his earlier defeat to Mendoza, on August 21, 1981, as part of the Gómez-Salvador Sánchez fight's undercard in Las Vegas.
- A knockout in nine rounds over Antonio Guido in Las Vegas as part of the Benítez-Durán title fight's undercard, January 30, 1982, after which the WBC ranked him as the number one challenger to Wilfredo Gómez's WBC world Super-Bantamweight title.
On March 27 of that year, Meza got his first world championship try, when he faced Gómez in Atlantic City, for the WBC world Super Bantamweight title. Meza lost to Gómez by a sixth-round knockout.
After losing to Gómez, Mesa took off one year from boxing. During this time, Gómez left the championship vacant; to challenge for the WBC's world Featherweight championship, and Jaime Garza won it by knocking out Bobby Berna for the vacant championship.
Meanwhile, Meza returned to boxing on April 26, 1983, knocking out Roberto Castillo in four rounds at Las Vegas.
He traveled to Thailand during July of that year, to fight Pongpan Sorphayathai on the 13th. of that month. He beat the Thai fighter by a third-round knockout.
On November 19, he fought Javier Barajas in Las Vegas, beating Barajas by a ten-round decision. The pair had a rematch, on May 16, 1984 in Las Vegas, and Meza won once again, by a ten-round decision. After that, he was ranked as the number one Super Bantamweight challenger once again by the WBC.
On November 3, he was given a second try at the championship when he faced Garza, in Kingston, New York. Meza became the first challenger in boxing history to be dropped in round one, get up and knock the world champion out in the same first round, when he beat Garza to become the WBC world Super Bantamweight champion that day.
On April 19, 1985 he and Julio César Chávez co-starred in an undercard, defending their world championships against Mike Ayala and Ruben Castillo, respectively, at the Inglewood Forum, in Inglewood, California. Both champions retained their titles with sixth-round knockout wins.
On August 18, he and Lupe Pintor squared off in a fight that garnered major interest among Mexican boxing fans. Meza lost his title in Tijuana that night, when Pintor dropped him twice, outpointing him over twelve rounds.
Pintor lost his title, in turn, to Samart Payakaroon, who gave Meza a chance to recover the WBC world Super Bantamweight championship on December 10, 1986 in Bangkok. Despite not fighting for more than a year, Meza lasted until the twelfth round with Payakaroon, who proceeded to knock him out in that round.
Meza had two fights in 1987, beating Lenny Valdez in the first round by knockout, and then losing to Javier Marquez by a knockout in round eight on October 30. After his fight with Marquez, he retired for ten years.
On February 14, 1997, he returned to boxing with a knockout victory, beating Esteban Lozoya in four rounds at Mexicali. But, in his next fight, he met fringe contender Wilfredo Negron of Puerto Rico, being knocked out by the Puerto Rican in the first round at Miami. Meza then retired for the second time.
He has stayed in retirement since. Because of his current age, it is doubtful he will ever box again as a professional.
His professional boxing record consisted of 45 wins and 9 losses, with 37 wins by knockout.
| WBC Super Bantamweight Champion
3 Nov 1984– 18 Aug 1985