Born in Budapest, Bogdán started his career with Hungarian Nemzeti Bajnokság I (Division One) team Vasas SC.

[9] On 29 August 2010, Bogdán made his Premier League debut when he replaced Jussi Jääskeläinen, who had been sent off for violent conduct, 37 minutes into the home game against Birmingham City.

[14] He performed well during Jääskeläinen's absence and when the Finn returned from injury, Bogdán kept his place for the home game against Arsenal on 1 February, in which he kept a clean sheet in a goalless draw.

[15][16] On 14 May 2012, Bolton Wanderers held their annual end of season dinner, with Bogdán the recipient of the main accolade as he was named the club's Player of the Year, which is voted by the supporters.

[17] Bogdán admitted he felt mixed emotions after getting the award[18] due to the draw against Stoke City which resulted in Bolton's relegation from the Premier League.

[22] On 20 December, he made his Premier League debut for the Reds in a 3–0 loss to Watford at Vicarage Road, dropping a corner kick after three minutes to allow Nathan Aké to open the scoring.

[23] Bogdán also played in a much-changed Liverpool side away to Exeter City on 8 January 2016 in the third round of the FA Cup, conceding direct from Lee Holmes' corner kick in a 2–2 draw.

[24] On the last match of the 2015–16 Premier League season, Bogdán was named in the starting line up against West Brom after a long absence from the first team squad.

In his final game, a 0–0 draw against Barnsley on 19 November, Bogdán suffered a tear in his anterior cruciate ligament and was substituted in the 59th minute.

[29] Bogdán played regularly for Hibernian during the early part of the 2018–19 season, but then missed several games due to concussion symptoms.