Rapport has been an active player since he started playing FIDE-rated events in 2006 when he was ten; in that year he won the U10 division of the 4th EU Youth Championship. In 2007, he took out the U12 Hungarian Boys Championship with 8.5/9, and was =4th in the Pannonia Cup Open A. In 2008, Rapport won First Saturday July IM Tournament, and came =5th in the Tenkes Kupa Open Championship. In 2009, he came second in the Genset Cup and in the First Saturday GM Tournament in August. In 2010, he came =2nd in the Marienbad Open 2010 - A - GM tournament and won the First Saturday GM February, defeating all four grandmasters who competed, thereby completing his third and final GM norm.

Since gaining his GM title, Rapport has finished equal second with Lajos Portisch behind Alexander Beliavsky in the Gotth' Art Cup GM Tournament in May 2010, scored 8.5/11 in the Hungarian Team Championship Final Szabo L, and scored 7/11 in the powerful Serbian League (2010), scoring +5 -3 =2 against the 10 grandmasters he encountered during the contest (the 11th player was an FM whom he also defeated). In October 2010, he scored 7.5/11 to come in 7th in the World Youth U14 championships, losing 22 ELO points in the process. However, he made up some ground in November when he scored 7.5/8 in the Hungarian Youth Team Championships, clawing back 10 ELO points. In 2011, he lost a match with Csaba Balogh 3.5-8 (+2 -3 =3), nonetheless gained Elo points for this result, and scored 5.5/10 at Tradewise Gibraltar (2011) and 7/11 at the 12th European Individual Championship (2011), gaining Elo points in each event. In May 2011, he scored 5.5/9 at the 41st International Tournament Bosna and in July, scored 7.5/10 at the Politiken Cup, a point off the lead. In November 2011, Rapport played a match with Austrian number 1 player Markus Ragger, losing by 3.5-4.5 (+1 -2 =5), nevertheless gaining 9 Elo as a result. Rapport finished 2011 with a solid 6/9 at Groningen and started 2012 with an equally solid 6/10 at Tradewise Gibraltar (2012), adding 16 points to his rating for the first bimonthly rating period of 2012. In early 2012, Rapport held his own in the 2011-12 Bundesliga and in the 13th European Individual Championship (2012) (5/10), but scored a strong 7/9 to place =2nd in the 2012 International Neckar Open. Subsequent strong results in the Hungarian Team Championship 2011-12 and in the Mitropa Cup 2012 have pushed his rating to over 2600 a few months after his 16th birthday. Later that year he was runner-up on tiebreak behind Turkish GM Alexander Ipatov in the World Junior Championship (2012).

2013 started auspiciously for Rapport. Competing at Tata Steel (Group B) (2013), he finished =1st on 9/13 with Arkadi Naiditsch, but won the event on tiebreak. It also added a massive 25 points to his rating almost bringing him into the top 100. The completion of the Austrian League has lifted Rapport into the top 100 for the first time. His excellent results in the Bundesliga and his =1st at the 2013 Neckar International Open added 20 points to his card to push him well up into the top 100. In May 2013, he won the 21st Sigeman & Co (2013), a category 15 event in Sweden, with 4.5/7 and in June 2013, was a member of the victorious Clichy team in the French TOP 12 competition.

With a standard rating of 2674, Rapport is the top ranked player U18 and the 4th ranked Junior (U20) in the world, and is also Hungary's number 5 player (June 2013). He is also number 69 in the world and his rapid rating is 2642 (world #87) and his blitz rating is 2598.