Zvjanginsev won the European under-16 championship in 1992. Two year later, he tied for first place in the Reykjavik Open with Hannes Stefánsson and Evgeny Pigusov.
In 1997, at the FIDE World Championship, which took place in Groningen, he single-handedly knocked out most of the U.S. contingent. In consecutive rounds, he defeated Joel Benjamin, Gregory Kaidanov and Yasser Seirawan, before losing to fellow Russian GM Alexey Dreev in round 4. In the same year Zvjanginsev won the Vidmar Memorial in Portorož.
In 2000, he was first at Essen (ahead of Dreev and Klaus Bischoff) and triumphed there again in 2002 (this time ahead of Leko). At the Mainz Chess Classic in 2003, he finished joint second behind Levon Aronian, repeating his placing the following year. At the Russian Championships of 2005, he took 3rd place at the Kazan qualifier and finished joint 4th at the Superfinal. In 2006, he tied for 2nd at the Poikovsky Karpov Tournament, behind Alexei Shirov.
In 2002, Zvjanginsev took part in the Russia vs Rest of the World match, held in Moscow.
He won the Russian Cup knockout tournament in 2011 by beating Denis Khismatullin in the final. In the 2013 Chigorin Memorial in Saint Petersburg, Zvjanginsev tied for 1st–11th with Pavel Eljanov, Dmitry Kokarev, Maxim Matlakov, Alexander Areshchenko, Denis Khismatullin, Oleg Korneev, Dragan Solak, Sanan Sjugirov, Ivan Bukavshin and Ildar Khairullin. In the 2016 Aeroflot Open, he tied 3rd-10th with Vladimir Fedoseev, Gata Kamsky, Daniil Dubov, Mateusz Bartel, Sanan Sjugirov, Maxim Matlakov, and Mikhail Kobalia.
In team competitions, he took team and individual silver medals at the 1997 European Team Chess Championship. At the 1994 Chess Olympiad, while still only an International Master, he helped the Russian second team obtain a team bronze medal. In the 1997 World Team Chess Championship Zvjanginsev won two gold medals, team and individual playing second reserve board. With the main Russian team, in 1998 and 2004, he contributed respectively to team gold and team silver medals at the Chess Olympiad
Wikipedia article: Vadim Zvjaginsev