In 1980 Adams was spotted as a promising young player; this led to European Junior Champion, Shaun Taulbut, being provided as a chess coach. Adams won the British Championship in 1989 at the age of seventeen. Making only rare appearances, he won again in 1997, jointly with Matthew Sadler and for a third time in 2010. Adams also won the British Rapidplay Championship in 1995, 1996 and 1999.

Two books co-written with his father, Bill Adams, Development of a Grandmaster (1991) and Chess in the Fast Lane (1996), discuss his early chess career.

Adams has performed strongly in a number of World Chess Championship tournaments.

In 1993 he finished equal first (with Viswanathan Anand) in the Groningen Interzonal tournament to determine challengers for the PCA World Chess Championship 1995. This took him to the Candidates Tournament matches, where he beat Sergei Tiviakov in the quarter finals, but lost to Anand in the semi-finals.

He also qualified for the Candidates Tournament for the FIDE World Chess Championship 1996, losing to Boris Gelfand in the first round of matches.

In 1997, he took part in the 1997-1998 FIDE World Championship, which, for the first time, was a large knock-out event, the winner of which would play a match against reigning champion, Anatoly Karpov. This tournament included most of the world's top players (Garry Kasparov, Vladimir Kramnik and Gata Kamsky were the only notable absentees), and Adams won short matches against Tamaz Giorgadze, Sergei Tiviakov, Peter Svidler, Loek van Wely, and Nigel Short, before coming up against Anand in the final round. Their four games at normal time controls were all drawn, as were four rapidplay games at quicker time limits, before Anand won the sudden-death game, thereby eliminating Adams from the competition.

The 1999 FIDE World Championship resulted in another semi-final finish for Adams, before losing to Vladimir Akopian. Yet again, he reached the semi-finals of the 2000 FIDE World Championship before losing to eventual winner Anand. In the 2002 FIDE World Championship he won his first three rounds before being knocked out in the 'round of 16' by Peter Svidler.

Adams came closest to claiming a world title at the 2004 FIDE Championship, when he reached the final, winning matches against Hussein Asabri, Karen Asrian, Hichem Hamdouchi, Hikaru Nakamura, Vladimir Akopian and Teimour Radjabov. However, he lost to Rustam Kasimdzhanov in the final (3½-4½ after rapidplay tie-breaks, the match having been tied 3-3 after the six standard games).

As runner-up in the 2004 event, Adams was one of eight players invited to the FIDE World Chess Championship 2005. He finished in equal sixth-seventh place, with a score of 5½ out of 14.

In May-June 2007, Adams participated in the Candidates Tournament to qualify for the FIDE World Chess Championship 2007. In the first round he drew 3-3 with Alexey Shirov, and was beaten 2½-0½ in the rapidplay playoff.

Among his other notable results are first at Terrassa in 1991, joint first at Dos Hermanas in 1995 (with Kamsky and Karpov), joint first at Dortmund in 1998 (with Kramnik and Svidler), and clear first at Dos Hermanas in 1999, ahead of Kramnik, Anand, Svidler, Karpov, Veselin Topalov, Judit Polgár and others.

In more recent times, he won the fifth Howard Staunton Memorial Tournament in August 2007, achieving a score of 8½/11 (six wins, five draws), picking up the top purse of £1000. Adams, the highest seeded player in the tournament by 45 Elo points, finished a full point ahead of Dutch Grand Masters Ivan Sokolov and Loek van Wely. The tournament, played in London, is an annual memorial to the British chess master Howard Staunton. Prior to the start of the tournament, on August 4, Adams married his longtime girlfriend, the actress Tara MacGowran, at a ceremony in Taunton, close to where they live.

In September 2007, Adams took part in a match between United Kingdom and China, held in Liverpool, England. Playing alongside Adams was former World Championship challenger GM Nigel Short. This chess event was the first time in almost 15 years that the two GMs had played chess together on British soil. Overall, he scored 3½/6, conceding one loss to GM Zhang Pengxiang (ELO 2649, at time of match) in round four. The UK team lost the match to China, 20-28, who had also defeated a Russian chess team a few weeks before.

Adams won the 2nd Ruy Lopez Masters tournament held in Mérida, Spain, 4—13 April 2008, scoring 5½/7 to finish a half point ahead of Zhang Pengxiang. The tournament was an eight player round-robin with an average rating of 2616 (FIDE category 15). In August 2008, there was a second successive victory in the Staunton Memorial. Adams finished on 8/11, ahead of Loek Van Wely (7½/11) and Jan Smeets (7/11). He followed this with a share of second place at the 4th EU Individual Open Chess Championship in Liverpool, joining Viktor Laznicka and Nigel Short on 7½/10 (after winner Jan Werle, 8/10).

He convincingly won the 2010 British Chess Championships, held in Canterbury, where he finished with an undefeated score of 9.5/11.

In June 2005, Adams took on an advanced chess super computer called Hydra in a six game match in London, England, with a prize of $10,000 for each draw and $25,000 for each win.

Hydra, housed in Abu Dhabi, at the time of the match consisted of 64 PCs each running 3.06 GHz Intel Xeon processors. Its designers said that it could, under optimal conditions, analyse up to 200 million positions a second, and, in the endgame, calculate up to 40 moves ahead.

Adams lost the match, drawing only the second game. The final score was Hydra 5½, Adams ½, with Adams only receiving $10,000 of the $145,000 prize fund.