Conti was born in Lecco, Lombardy. In 2002, he had a successful trial at A.C. Milan yet refused to join the team due to his dislike of the club's youth training center in Linate area (or simply its far-from-home location). He also got an offer from Inter Milan but eventually joined Atalanta's youth academy where he played alongside such players as Mattia Caldara and Roberto Gagliardini for over a decade.
In July 2013, Conti was loaned to Lega Pro side Perugia. He made his professional debut on 4 August 2013, starting in a 0–1 Coppa Italia home loss against Savona. His first league appearance occurred later that month, a 2–2 away draw against Nocerina. Conti made 16 league appearances, helping his team achieve promotion to Serie B.
On 10 July 2014, Conti moved to a fellow second-tier team Virtus Lanciano, in a season-long loan deal. Conti made 24 league appearances while his team finished 14th out of 22. In the summer of 2015, he returned to Atalanta and was included in the first-team squad for the upcoming Serie A season.
On 30 September 2015, Conti signed a new contract with Atalanta until July 2019. He made his debut for the club on 2 December, in a 1–3 loss against Udinese in Coppa Italia. Conti made his debut in Serie A on 6 January 2016, in a 1–2 loss against the same opponent. His first goal in the competition occurred on 3 February, in a 2–1 win against Verona.
Conti managed to score 8 goals in 32 appearances during 2016–17 Serie A season, setting a personal record for scoring. His main playing position was right wing-back in a 3–4–3 line-up.
On 7 July 2017, Conti joined Milan in exchange for a reported fee of €24 million, plus Matteo Pessina. He made his debut for the club on 27 July 2017, replacing Ignazio Abate as a second-half substitute away to CS U Craiova. He made his league debut in a the 3–0 win against Crotone. On 15 September 2017, he injured his ACL in training. The club reported that his recovery time is expected to be six months.
On 12 August 2015, Conti made his debut with the Italy U21 team, in a friendly match against Hungary.
In June 2017, he was included in the Italy under-21 squad for the 2017 UEFA European Under-21 Championship by manager Luigi Di Biagio.
In Auguest 2017, he was called up to the senior squad by Gian Piero Ventura 2018 World Cup qualification match against Spain on 2 September and, on 5 September against Israel. He made his senior international debut in Italy's 1–0 home win over Israel, but was forced off early due to injury and replaced by Davide Zappacosta.
Conti can play as a right back or a right wing-back, although he is perhaps more comfortable in the latter position. Gifted with great pace he thrives going forward and making overlapping runs. He excels at staying wide and crossing but also has a knack for getting into the box and scoring himself. His great stamina allows him to make attacking runs and rapidly get back into position when needed.As of 20 August 2017
As of 5 September 2017