He started playing for Betis, making his debut with the first team in 2012.
Born in Seville, Andalusia, Adrián played as a striker and a winger for CD Altair until the age of 10, when the previous goalkeeper left and he changed position to that of goalkeeper. He later signed for Real Betis, where he finished his formation.
Adrián spent his first two seasons with the C-team, and a further five with the reserves, four of those in Segunda División B, successively backing up René and Brimah Razak. He was also loaned to CD Alcalá, in 2008 and CD Utrera the following year. He was promoted to the first team for 2011–12 as third-choice – still continuing to appear for the B's – but suffered a severe injury to his cruciate ligaments in November that kept him out of action for five-and-a-half months.
Adrián made his La Liga debut on 29 September 2012, in a 0–4 away loss against Málaga CF: after Casto was sent off in the early minutes, he replaced out-field player Salvador Agra and went on to be named Man of the Match. Still in the 2012–13 campaign he became the starter, namely putting on a Man of the match performance in a 1–0 home win against Real Madrid on 24 November. Following his debut he went on to start the next 31 games, keeping eleven clean sheets as the club finished in seventh place and qualified to the UEFA Europa League.
Having been watched playing for Betis by West Ham United manager Sam Allardyce and goalkeeping coach Martyn Margetson, Adrián was persuaded to join the club. On 5 June 2013, it was announced that he would sign a three-year contract on 1 July, with the Premier League club having the option to extend this deal for a further two seasons.
Adrián made his debut for the Hammers on 27 August 2013, in a 2–1 home win against Cheltenham Town for the season's League Cup: during the game he fouled Jermaine McGlashan to concede a penalty, which Matt Richards converted. His first league appearance came on 21 December, a 1–3 loss at Manchester United.
On 11 January 2014 Adrián kept his first Premier League clean sheet, in the 2–0 away win over Cardiff City. On 6 May he attended the club's awards and won 'Save of the Season' for a fingertip save from Oscar against Chelsea in January. He also won 'Best Individual Performance' for the same game, in which he kept a clean sheet in a goalless draw, as well as 'Signing of the Season'; he was also named runner-up 'Hammer of the Year' to Mark Noble. By the end of the 2013–14 season he had become West Ham's first-choice, replacing previous incumbent Jussi Jääskeläinen.
In an FA Cup third-round replay penalty shootout against Everton on 13 January 2015, Adrián saved from Steven Naismith. The shootout went to sudden death, in which he scored the winner after his Everton counterpart, compatriot Joel, struck the crossbar. Despite having never taken a competitive penalty before, he was so confident that he would score and win the game that he removed his goalkeeping gloves before taking it – his team won it 9–8.
On 11 February 2015, Adrián was given a straight red card in a goalless draw at Southampton for handling the ball outside the penalty area while pressured by Sadio Mané. His suspension for this dismissal was rescinded by the FA, although West Ham were charged with failing to control their players following the incident, being fined £30,000. He played all 38 league games and all four FA Cup matches during the campaign.
Adrián received a straight red card in added time at the end of West Ham's 1–2 home loss to Leicester City on 15 August 2015 for a high challenge on Jamie Vardy, after going up for a corner kick. As the team had already made all of their substitutions, Carl Jenkinson became the goalkeeper for the remainder of the match. In October 2015 he signed a new two-year contract, with the option of a further two years, keeping him at the club until 2017.
On 26 August 2016, Adrián received his first call-up to the senior Spain squad by new manager Julen Lopetegui for matches against Belgium and Liechtenstein.As of 23 August 2017