[4] He made his senior debut for Leinster in a Magners League fixture against Scarlets on 20 February 2010, coming on as a late substitution and setting up Johnny Sexton's try.
He started in Munster's first European Rugby Champions Cup game against Sale Sharks on 18 October 2014.
[16] Conway started at fullback in Munster's opening 2015–16 European Rugby Champions Cup pool game against Benetton on 14 November 2015.
[18] On 1 April 2017, Conway scored a try in Munster's 41–16 2016–17 European Rugby Champions Cup quarter-final win against Toulouse.
[23] Conway scored a 74th minute try, converted by Ian Keatley, to earn Munster a 20–19 win against Toulon in the quarter-finals of the 2017–18 European Rugby Champions Cup.
[47] On 18 March, Conway made his debut for the senior Ireland team, coming on as a half-time replacement for Munster teammate Keith Earls in the sides 13–9 victory against England in the Aviva Stadium.
[50][51] Having been selected in the squad for the 2017 Autumn Internationals,[52] Conway scored his first try for Ireland in their 38–3 win against South Africa on 11 November 2017.
[56] Conway scored a hat-trick and earned the Man-of-the-Match award in Ireland's 57–14 win against the United States during the 2018 Autumn Tests on 24 November 2018.
[59] He was selected in the 31-man Ireland squad for the 2019 Rugby World Cup,[60] having featured in the warm-up matches against Italy,[61] England[62] and Wales.
[66] Retained by new head coach Andy Farrell in his squad for the 2020 Six Nations Championship, Conway in Ireland's 19–12 opening win against Scotland on 1 February 2020,[67] their 24–14 win against defending champions Wales on 8 February,[68] and their 24–12 defeat against England,[69] before the tournament was suspended due to the COVID-19 pandemic.
[73] With the usual format of end-of-year international tests not possible due to the COVID-19 pandemic, Ireland instead participated in the Autumn Nations Cup.
[82] That knee injury later required surgery, which ruled Conway out of the 2022 Ireland rugby union tour of New Zealand.
[83] Conway and his partner Lizzie welcomed their first child, a daughter, in April 2022,[84] before the couple married in August of that year.