[1][2] In March 2020, the International Cricket Council (ICC) confirmed that Conway, who was born in South Africa, would be eligible to play for New Zealand from 28 August 2020.

[8] During the same month, Conway was also a member of the New Zealand team that won the 2021 ICC World Test Championship final, and had a key contribution scoring 54 runs in the first innings.

[14] In the second round of the 2018–19 Plunket Shield season, Conway scored an unbeaten double century against Otago at the Basin Reserve.

[24] In April 2020, Conway was named the men's domestic player of the year by New Zealand Cricket at their annual award ceremony.

[29] In the 2023 IPL season, CSK would go on to win the title, beating the Gujarat Titans in the final, where Conway played a pivotal knock of 47(25) in the chase and was declared the player of the match.

[31] In June 2023, Texas Super Kings added Conway to the roster for the inaugural season of Major League Cricket.

[36] In December 2020, Conway was added to New Zealand's Test squad,[37] covering for BJ Watling who suffered a hamstring injury.

[47] At the age of 29 years and 329 days, Conway also became the oldest batsman to score a double century on his Test debut.

[49] However, Conway was ruled out of New Zealand's squad for the final of the tournament after breaking his hand during the semi-final match against England.