Son of legendary City keeper Bryan Gunn, Angus played for Norwich's Under-15s in their victorious Eendracht Aalst Tournament campaign in May 2010. He was then called up to the England Under-16s for the Victory Shield matches in October 2010

Angus made his first start for Norwich's Under-18s in their 1-1 draw with Crystal Palace on 5th February 2011

Gunn made his debut for England's Under-16s in a friendly against Slovenia on 16th February 2011, keeping a clean sheet before being substituted at half-time.

When his father relocated to the North West in May 2011, Angus followed. Bryan explained the switch away from Norfolk, with him getting a new role with a sports agency. “This is a good opportunity to get back into the sporting arena. For Susan it is also an opportunity to extend her art work to Manchester, where she has exhibited in the past. Our daughter Melissa is also going to Manchester University. Angus, as a 15-year-old, will hopefully come and we’ll find a good school where he can finish off his GCSEs and a football club which can further help develop his skills.” It was announced at the end of June 2011, that Angus had joined Man City's youth set-up.

Angus made his debut for Man City's Under-18s against Marseille in the Next Gen series on October 6th in France, losing 3-0 as usual keeper Eirik Johansen was away on international duty with Norway Under-21s

A tribunal ruled in December 2011 that Man City were to pay Norwich £250,000 for Gunn, although the Canaries were asking for £1million. Norwich will however get add-ons dependent on Gunn becoming a first-team player at the Etihad and ultimately succeeding England number one Joe Hart.

In October 2013, Gunn followed in his father's footsteps with a victory over Bayern Munich; Gunn kept goal on 2nd October 2013 as Man City beat Bayern 6-0 in the UEFA Youth League.

In June 2015, he was linked with a loan move to his father's old club Aberdeen. He signed a new four year deal with Manchester City on 7th July 2015, after helping City's development squad win the Premier League International Cup and reach the quarter finals of the UEFA Youth League.

Gunn made his England Under-21 debut on 3rd September 2015 coming on as substitute in the 1-0 win over the USA Under-23 squad.

Angus returned to Norwich on 6th June 2017 signing on a season long loan from Man City. He is yet to make a senior appearance for Man City, but was regularly an unused substitute for the Premier League side last season. He was part of the England Under-21 squad that reached the semi-finals of the UEFA Under-21 Championships although he didn't play.

Gunn made his Norwich debut in the 1-1 draw away to Fulham on the opening day of the 2017/18 season (5th August 2017) - this was his first ever first team appearance at any level. His home debut came three days later in the 3-2 win over Swindon in the League Cup. He got his first professional football clean sheet on 16 August 2017, in a 2–0 home win over QPR.

After impressing for England Under-21's, including keeping a clean sheet in a 2-0 win in Ukraine on 10th November 2017, he was called up to the full England squad for the friendly against Brazil on 14th November 2017 following an injury to Jack Butland. Fellow Under-21 teammates Dominic Solanke and Lewis Cook were also promoted. Gunn was on the bench for the Brazil game but Joe Hart played all 90 minutes in a 0-0 draw.

Gunn ended the 2017/18 season with 51 appearances for Norwich.

On 27th June 2018, he was linked with a £15million move to Southampton. Leicester were also reported to be interested whilst other reports suggested that Man City would loan him to Leeds United. On 10th July 2018, Gunn completed a £13.5million move to Southampton as he was behind Ederson and Claudio Bravo in the fight for a first-team place at Man City. The deal would see an initial payment of £10m for the Norwich-born 22-year-old, with up to £3.5m in add-ons. It is unknown what this will mean for existing Southampton keepers Fraser Forster and Alex McCarthy. Gunn made his Premier League debut in Southampton's 0-0 draw with Southampton on 2nd January 2019, winning the Man of the Match award. At 22 years and 346 days, Gunn is the youngest English goalkeeper to keep a clean sheet on his Premier League debut since Ben Amos for Manchester United against Stoke in January 2012 (21 years, 295 days).

In October 2019 it was reported that Scotland boss Steve Clarke had failed in an attempt to persude Angus to switch allegience and play for Scotland like his father.

Falling behind Alex McCarthy and Fraser Forster at Southampton following the 9-0 defeat to Leicester in October 2019 and he was loaned to Stoke City on 16th October 2020 replacing former England number 1 Jack Butland who had joined Stoke City. Gunn will compete for the goalkeeper's position with Adam Davies, who has kept eight clean sheets in 12 matches since replacing Butland as first choice in July 2020.

Reports in early June 2021 suggested Southampton had agreed to sell Gunn to Norwich for £5 million plus incentives; however Leeds were also reported to be interested on 12th June. Gunn would be undeerstudy to Tim Krul with the view of replacing him in two to three seasons. This was finalised on 23rd June 2021. He had played 30 games for Southampton. The move was reported to be an initial £2.5m with add-ons potentially taking the deal closer to £5m. A series of ambitious performance-based could see the fee rise to £10m, but that would be dependant on Gunn achieving a series of things during his time at the club.. Gunn signed a contract until 2025.

Head coach Daniel Farke added: “We got the feeling that Angus will be the perfect fit to our goalkeeping group. We’ve got two very experienced goalkeepers in Tim Krul and Michael McGovern and a young goalkeeper in Daniel Barden. Angus fits perfectly into this goalkeeping team. He’s in a perfect age and is full of potential. He’s proven that he can play on the top level and was fantastic for us three seasons ago. He played almost each and every second for us and of course he has a huge identification and connection with the football club through his father. He has had a call-up to the England national team and there was a big money move to Southampton. That says a lot about his potential. After he was our number one goalkeeper three years ago we got the feeling that we helped give him the perfect platform. We needed a strong addition to our goalkeeping squad and for the short, medium and long term we believe that Angus is the perfect fit.”

He made 11 appearances during the 2021/22 season which saw Norwich relegated from the top flight.

In January 2023 it was reported that Scotland were considering asking Gunn to switch his allegience to Scotland after injuries to number one Criag Gordon leaving the Scots with untested deputies Jon McLaughlin, Liam Kelly and Zander Clarke. Former Scotland boss Alex McLeish revealed that he twice approached Gunn about joining the Scotland side.

In February 2023 he was linked with a move to Rangers. Supposedly multiple senior officials at Ibrox admire what Gunn can offer in Glasgow. The Premiership giants believe the shot-stopper can be a real asset for the club going forward and see him as their long term number one. Norwich allegedly have slapped a £3million price tag on Gunn.

Gunn declared for Scotland on 12th March 2023 and was called up for Scotland's European Championship qualifiers against Cyprus and Spain along with teammates Grant Hanley and Kenny McLean. Gunn made his debut on 25th March 2023 playing the full 90 minutes and keeping a clean sheet as Scotland defeated Cyprus 3-0.

He made 32 appearances for Norwich during the 2022/23 season.