Kveinys was awarded the Soviet Master of Sport title for chess in 1981. He won the Lithuanian championship 5 times: in 1983 (shared with Eduardas Rozentalis),[2] 1986,[3] 2001 (shared with Šarūnas Šulskis), 2008 and 2012.[4] In 2010 he won a tournament in Moss, Norway.[5] In March 2013 he won the Mallorca Masters.
In 2015, Kveinys tied for first place in the U.S. Masters Chess Championship. In 2016, he won the Chess Festival in Bad Wörishofen.[6] He was awarded the FIDE titles of International Master in 1990 and Grandmaster in 1992. He was a graduate of the chess department of the Moscow Institute of Physical Culture and Sports (now Russian State University of Physical Education, Sport, Youth and Tourism).[7]
Kveinys played for Lithuania in 8 Chess Olympiads:[8]
He has also represented Lithuania in 5 European Team Chess Championships:[9]
- In 1992, at second board in the 10th European Team Chess Championship in Debrecen (+2 −2 =5);
- In 2003, at first board in the 14th European Team Chess Championship in Plovdiv (+2 −2 =4);
- In 2005, at third board in the 15th European Team Chess Championship in Gothenburg (+1 −0 =8);
- In 2007, at third board in the 16th European Team Chess Championship in Heraklion (+3 −3 =2);
- In 2015, at first board in the 19th European Team Chess Championship in Warsaw (+3 −4 =2).
