A Glaswegian born in 1994, Robertson was on Celtic's books as a schoolboy but released when he was fifteen. He went on to join Scottish League Two amateurs Queens Park, forcing his way into their first team for the 2012/13 season before being signed by Dundee United. After an impressive season, he was named PFA young player of the year in Scotland, having scored three goals from the left-back position. He also made his Scotland debut whilst still a teenager in March 2014, coming on as a second-half substitute against Poland in Warsaw. In the summer of 2014, Robertson joined Hull City where Robertson had an eventful three years. He was twice relegated from the Premier League as well as being promoted from the Championship. He made 99 league appearances, scoring three times and continued to be called up by Scotland, scoring his first international goal against England in a friendly at Celtic Park in November 2014.

Robertson joined Liverpool on 21 July 2017 in a deal reported as being worth as much as £10million. He expressed a desire to prove he can perform at the highest level and Jurgen Klopp told reporters he believed it was the perfect move for him. He said " For Andrew, this is another big step on what has been quite an incredible personal journey in a very short space of time. I know our environment will benefit him and help him push himself even more than he has already. This is a player who does not limit his ambition."

In his first season at Anfield, Robertson had to bide his time, but he worked hard in training and took his chances when they came. A strong tackler who chased down every opponent and could put in a decent cross, had by the end of the campaign ousted Alberto Moreno from the left-back spot and scored his first goal for the club in the last league game of the season against Brighton. Robertson began the 2018/19 season as first-choice left-back and when he joined up with the Scottish international squad at the start of September, he was announced as the new national captain. Robertson continued to exceed all expectations. He appeared in 36 of the 38 Premier League games and missed just one game in the Champions League, through suspension. Retaining his fitness, the only time he was laid up was when dental surgery ruled him out of a Scottish World Cup qualifier in Kazakhstan. Robertson defended with tenacity, terrorised opposition defenders with his marauding runs down the flank, whipped in crosses and assisted with eleven goals on the home front, a joint Premier League record for a defender. He signed a new and improved contract in January 2019 but remained grounded, tweeting after the Champions League triumph against Tottenham: "Just a wee guy from Glasgow living he absolute dream."

Robertson embodied Liverpool's desire to win their first league title for 30 years with his performances in the 2019/20 season. He was absolutely relentless and his opponents undoubtedly felt his wrath. In May 2019 Robertson wrote a heartfelt letter on the Players' Tribune. It was an astounding mature message that revealed the sacrifices he had made to get to the very top of world football. In conclusion he said: "And if there’s one thing I can guarantee about this team, about this group of players, it’s that we will stop at nothing to try to make our supporters’ dream come true. If that does happen, it won’t be a fairy tale. It’ll be because we deserve it." Liverpool certainly deserved the Premier League title, finishing on 99 points, 18 ahead of their closest rivals Manchester City and with the trophy in the bag when seven rounds were left to play of the competition. Robertson is worth his weight in gold and to see him win the greatest trophies on offer is truly heartwarming.

During 2020/21 Robertson was one of the few players whose season was not blighted by injury. He started every single one of Liverpool's 48 Premier League and Champions League games, being substituted in only five of them. His presence was essential to help those untried centre backs who had to come in the side due to a succession of injuries in that position, most importantly to Virgil van Dijk. In the 2021/22 season, Robertson faced increased competition for his place due to the rise of Kostas Tsimikas, yet he responded with characteristic professionalism, consistently delivering high-level performances in the Premier League, FA Cup, and European competitions. His stamina, overlapping runs, and precise crossing continued to be key elements of Liverpool’s attacking play and he achieved a personal best of 15 assists during this successful campaign where two cups were won and were agonisingly close to the Champions League and Premier League titles. He even scored three league goals including the opener in a 2-0 win over Everton at Anfield.

Robertson was made an MBE in the New Year Honours list in January 2023 for services to association football, charity and young people. On December 26, 2022, in a 3–1 victory over Aston Villa, Robertson provided his fifth assist of the season, taking his career total to 54. This achievement surpassed the previous record for a defender in the Premier League held by Leighton Baines, who had 53 assists. Notably, Robertson accomplished this feat in 189 fewer appearances than Baines. The following season saw Robertson appear 23 times in the Premier League, scoring three goals. He had a frustrating three month spell on the sidelines between October and January due to a shoulder injury. In Euro 2024 Robertson captained Scotland but suffered the disappointment of finishing bottom of their group.

During the victorious 2024/25 season, Robertson experienced a noticeable dip in form, leading to increased scrutiny and speculation about his future at Liverpool. Uncharacteristic mistakes crept into his performances, such as misjudged clearances or misplaced passes that allowed opponents to create scoring opportunities. This period of uncertainty coincided with Liverpool's acquisition of Milos Kerkez from Bournemouth in the summer of 2025, signalling a potential shift in the club's left-back options. Reports indicated that Atlético Madrid had identified Robertson as a top target, with manager Diego Simeone reportedly keen on bringing the experienced defender to La Liga. Atlético Madrid decided to pursue other options and Robertson stayed on, still a vital player for the left-back position as first-choice Kerkez is significantly less experienced. Robertson’s leadership qualities has gained him a lot of respect within the Liverpool squad and ahead of the 2024/25 season, he was officially named Liverpool’s vice-captain. In this role, Robertson not only continued to deliver on the pitch but also mentoring emerging talents while maintaining the high standards expected at Liverpool.