In 2012, he signed for Eintracht Frankfurt for a fee of €1.5 million and played regularly over three years, also competing in the UEFA Europa League.

Born in Merzig, West Germany, Trapp began his football career at FC Brotdorf, joining at seven years old.

[10] Trapp made his first appearance in the first team on 9 August 2008 in the 2008–09 DFB-Pokal round 1 against FC Carl Zeiss Jena, losing 2–1.

[12] Trapp made another appearance in the first team ten days later on 31 July 2009 in the 2009–10 DFB-Pokal round 1 against Eintracht Braunschweig, where he kept a clean sheet, in a 1–0.

[17] On 12 March 2011, with regular goalkeeper Tobias Sippel sidelined with influenza, he made his Bundesliga debut in a 2–1 home win over SC Freiburg.

[22] His performance was praised by Manager Marco Kurz, who said: "Kevin once again proved his class in a very emotional, intense game and saved the point.

On 7 May 2012, Trapp left relegated Kaiserslautern and joined promoted Eintracht Frankfurt on a four-year contract for a fee of €1.5 million.

[29] Trapp made his Eintracht Frankfurt debut, where he started the match, only to be sent–off in the 19th minute for a professional foul on Jakub Sylvestr, as the club lost 3–0 against Erzgebirge Aue in the first round of the DFB–Pokal.

[30] Despite this, Trapp made his league debut for the club, helping the side win 2–1 against Bayer Leverkusen in the opening game of the season.

[33] The team had a good start to his first season, sitting in third place by November, and he received plaudits for his performances against leaders Bayern Munich despite losing 2–0.

[40] Trapp played in both legs of the UEFA Europa League Play–Offs against Qarabağ, helping the side win 4–1 on aggregate to qualify the Group Stage.

Ahead of the 2014–15 season, Trapp was linked with a move to Borussia Mönchengladbach as a replacement for Marc-André ter Stegen, but stayed at Eintracht Frankfurt.

[53] Since returning from injury, Trapp continued to regain his captaincy and his role as the club's first choice goalkeeper for the rest of the 2014–15 season.

On 8 July 2015,[55] Trapp signed a five-year deal with Ligue 1 champions Paris Saint-Germain for a fee of around €10 million, and was assigned the number 16 jersey.

[56] Trapp made his debut on 1 August in the 2015 Trophée des Champions in Montreal, keeping a clean sheet in a 2–0 win over Lyon.

[62] Trapp went on to earn a total of four clean sheets, as PSG advanced to the UEFA Champions League Group Stage.

At the start of the 2016–17 season, Trapp was in goal as PSG won the 2016 Trophée des Champions in Austria, a 4–1 win on 6 August against Lyon again.

[70] But following Areola's poor performance, he returned to the starting line–up once again, coming against FC Lorient on 21 December 2016 and kept a clean sheet, as PSG won 5–0.

[74] In a follow–up against Barcelona in the first leg of UEFA Champions League Round of 16, Trapp helped the side win 4–0, earning himself a clean sheet.

[84] Trapp then helped PSG reach the final of Coupe de la Ligue after beating Rennes 3–2 in the semi–finals on 30 January 2018.

[85] A week later on 7 February 2018, he was sent–off for receiving a straight red card in a last minutes, in a 4–1 win against Sochaux in the third round of the Coupe de France.

[87] On 31 March 2018, in a repeat of the previous season's final, Trapp started in goal as PSG retained the Coupe de la Ligue title with a 3–0 win over Monaco.

[94] Following a 2–1 loss against VfL Wolfsburg on 2 December 2018, Trapp made a comment about Sascha Stegemann, resulting in him being fined 8,000 euros.

[95] Throughout March, he kept five clean sheets in all competitions, including a 2–0 aggregate win against Inter Milan in the UEFA Europa League Round of 16.

[96] Trapp then played in both legs of the UEFA Europa League semi–finals against Chelsea; although he saved one penalty from César Azpilicueta in the shootout, they lost 5–4 as a result.

[97] After a successful loan spell, Trapp re-signed with Eintracht Frankfurt on 7 August 2019 on a five-year deal worth €7 million.

[99] Trapp then played in both legs of the UEFA Europa League Play–Off Round against RC Strasbourg, as he helped the side win 3–1 on aggregate to send them through to the Group Stage.

[114] He made his Germany U21 debut on 7 September 2010, starting the whole game, in a 3–0 win against Northern Ireland U21 and keeping a clean sheet in a process.

[115] Trapp then kept three consecutive clean sheets during the UEFA European Under-21 Championship qualification, coming against San Marino U21, Belarus U21 and Bosnia and Herzegovina U21.

[122] On 4 June 2018, Trapp was selected in Germany's final 23-man squad for the 2018 FIFA World Cup as a third-choice goalkeeper behind Manuel Neuer and Marc-André ter Stegen.