He attended Károly Zipernovszky Secondary School of Mechanical Engineering in Pécs, where he obtained a qualification of electrical technician in 1979.

Varju was a member of the General Assembly of Pest County between 1994 and 2002, where he worked for the youth and sport committee.

Varju served as Secretary of State for National Development and Economy in the cabinets of Ferenc Gyurcsány and Gordon Bajnai, from 1 December 2008 to 29 May 2010.

After the 2010 national election, when the Socialist Party was defeated by Fidesz, Varju was a member of the Health Committee from 2010 to 2011.

When the platform split from the Socialists on 22 October 2011 he joined to the newly formed party and left the MSZP and its parliamentary group.

[4] In the 2021 Hungarian opposition primaries, his party also ran in Budapest 11 - Újpest, Angyalföld - constituency with the support of the All Hungary Movement.

With the support of the opposition parties DK, MSZP, Momentum, Párbeszéd, LMP, Jobbik and the Liberal Party, László Varju won the primary election and the 2022 Hungarian parliamentary election in his constituency as the sole candidate.