When Zampa was in school, he used to bowl medium pace but after therelease of Cricket Australia's new guidelines for Under-14 matches, in which they had put restriction on the speed of fast bowlers, he started bowling leg spin. In 2009, he was selected in the Under-19 team and represented Australia in the their successful 2010 Under-19 World Cup campaign.
With his old-school technique of tossing the ball above the eye-line, he was highly successful in his youth career. As a result, New South Wales signed him in 2010 but he didn't get a chance to play instantly as they already had a couple of successful spinners. Zampa made his first class debut in the 2012-13 season of Sheffield Shield against Queensland and took 5 wickets in the match. In 2013, he was signed by South Australia and took his first hat-trick in a match of 2014-15 Matador BBQs One-Day Cup against Western Australia. In Big Bash, he has featured for Adelaide Strikers and Melbourne Stars. So far, Zampa has played 38 First-Class and 99 List-A games, in which he has taken 105 and 147 wickets respectively.
Zampa's consistent economical bowling in T20s caught eyes of several franchises owners and
was consequently bought by the now-defunct Rising Pune Supergiants. Playing under the guidance of MS Dhoni and Steve Smith, Zampa achieved the second best bowling figure in the IPL history after taking 6 scalps for 19 runs against Sunrisers Hyderabad.
This wrist-spinner seemed to have an affinity towards the turning Indian pitches as he took 19 wickets in 11 IPL games. However, after two relatively good seasons in IPL, he wasn't picked by any franchise but in 2020 Zampa was picked as a replacement for countryman Kane Richardson in Royal Challengers Bangalore ahead of the 13th edition of the league.
In 2015, Zampa was added in the Australian team for 2015-16 Chappell-Hadlee Trophy against New Zealand and made his debut in the second match of the series on 6 February 2016. In his debut match, he bowled his full quota of 10 overs and took 2 scalps for 57 runs. Zampa was in Australia's squad for the 2017 ICC Champions Trophy, but he didn't get many opportunities to play when two of Australia's three group stage matches were washed out and Australia was unable to progress to the finals. In April 2019, he was named in Australia's squad for the 2019 Cricket World Cup.Till date, Zampa has played 58 ODIs and has 85 wickets in his name.
On 4 March 2016, Zampa debuted in the shortest format of the game against South Africa. However, he didn't get any wicket in the match but bowled his quota with a decent economy of 6.50. Since debut, Zampa performed well enough to maintain his place in the team. Especially in T20s, he emerged as an economical wicket-taker. Consequently, his name was added in the team for the 2016 T20 World Cup. Though, Australia couldn't do well in the tournament but Zampa finished as Australia's highest wicket-taker at the excellent economy rate of 6.27.
Despite of his consistency, he was dropped for the second game of the 2016-17 three-match T20 series against Sri Lanka which led Australia to lose the match and the series.He then played in the dead-rubber and took 3 wickets for 25. Consequently Australia won the match. In T20Is, he has 36 wickets in 33 fixtures.
Zampa belongs to New South Wales, but in 2013, he shifted to South Australia to play his domestic cricket. In honour of his mate Late Phil Hughes, he wears #63 jersey. He worked as a Dolphin Trainer at SeaWorld Entertainment. His father Darren played cricket for New South Wales. He is a dog lover and has a pet dog, Zaidee. Zampa is engaged to Harriet Palmer.