Born in Leganés, Madrid, Robles is a youth product of Getafe, joining neighbouring Atlético Madrid in 2005 to finish his development. He spent his first professional season, (2009–10), with the reserve side in Segunda División B and, on 27 December 2009, signed his first professional contract, running until June 2014.

On 8 May 2010, Robles made his La Liga debut in a 1–1 away draw against Sporting de Gijón: starter David de Gea (another Atlético youth graduate) was being rested for the UEFA Europa League final, and his replacement, Sergio Asenjo, suffered a severe knee injury after only ten minutes. As a result of Asenjo’s injury Robles took his place on the bench for the UEFA Europa League final as Atletico defeated Fulham.

In the 2010–11 campaign, Robles acted mostly as backup, particularly after Asenjo left in January 2011 to join Málaga on loan, only to be again seriously injured in the process. After De Gea was signed by Manchester United in the off-season, Robles was made first-choice by newly appointed manager Gregorio Manzano, but he lost his position after only three official games to the Belgian Thibaut Courtois, on loan from Chelsea.

On 31 January 2012, Robles was loaned to neighbouring Rayo Vallecano – also in the top division – who had lost first-choice keeper Dani Giménez to injury. He played his first game five days later in a 2–1 away win against Real Zaragoza.

Robles was loaned again in the 2013 winter transfer window, joining a host of compatriots – including manager Roberto Martínez – at Premier League club Wigan Athletic. He made his debut on 26 January, keeping a clean sheet in a 1–0 win over non-league side Macclesfield Town in the fourth round of the FA Cup.

Robles made his first league appearance on 17 March 2013, starting in a 2–1 home victory against Newcastle United. He soon won the battle for first-choice status over Ali Al-Habsi.

On 11 May 2013, Robles started in Wigan’s first-ever FA Cup final appearance, preventing Carlos Tevez from scoring twice in the first 45 minutes and eventually keeping a clean sheet in an historic 1–0 win over Manchester City at Wembley.

Robles signed for Everton on 9 July 2013, re-joining Martínez on a five-year contract for an undisclosed fee. On 26 December 2013, he played in the Toffees’ home defeat against Sunderland. Robles came on as a 24th-minute substitute for Leon Osman, as starting goalkeeper Tim Howard had been sent off for a professional foul. Robles was unable to save the resulting penalty, however, as Everton lost the game 1–0. Robles then made his first Premier League start for Everton against Southampton three days later due to Howard’s suspension, with Everton winning 2–1.

On 13 January 2015, in a penalty shootout at the end of an FA Cup third round replay against West Ham United, Robles saved Stewart Downing’s penalty kick. When the shootout went to sudden death, however, Robles hit the crossbar while West Ham’s goalkeeper, his compatriot Adrián, scored to eliminate Everton. Martínez assigned Robles as be the first-choice goalkeeper, ahead of Howard, in February 2016.

After a match against Burnley in which his gaffe resulted in a penalty, Robles was dropped by Everton manager Ronald Koeman for the away trip to West Ham on 22 April 2017 in a Premier League fixture after claiming the first team spot back in December 2016. As Koeman was not convinced by Robles, Everton completed the signing of Jordan Pickford from Sunderland in June 2017, putting Robles further down in the pecking order. His contract was allowed to expire at the end of June 2018.

On 5 July 2018, Robles signed a four-year contract with Real Betis. He was first-choice for cup and European matches, while Pau López started league games. As López left for Roma in the next year, he was the first-choice for league matches.