Veselý has been playing tennis since he was five and can look back on a very successful junior career. With Oliver Golding he won the gold medal in doubles at the 2010 Summer Youth Olympic Games in Singapore, and in singles he reached the second round. In 2011 he won the Australian Open in singles and doubles and was also in the finals of the US Open in both competitions. In 2011 Veselý reached another final at Wimbledon . He also led the world rankings for juniors for a long time.
In 2009 he started the first tournaments on the ITF Future Tour , where he has won nine singles and five doubles titles so far. He won his first title on the ATP Challenger Tour on April 14, 2013 with a two-set victory over the German Simon Greul in Mersin . A month later, his second Challenger title followed when he defeated Steve Darcis in the final of the Prosperita Open in Ostrava .
Veselý was coached by several coaches during his career. In 2019 - after injuries and a form low - the current captain of the Czech Davis Cup team Jaroslav Navrátil is his coach.
Davis Cup
Veselý played his first game for the Czech Davis Cup team in February 2013 against Switzerland . He lost the no longer decisive individual against Henri Laaksonen with 6: 0, 3: 6 and 1: 6. In the semi-final against Argentina , he also lost the last encounter against Leonardo Mayer in straight sets. For the encounter in the final against Serbia , which won the Czech selection 3-2, he was no longer in the squad.