Wales were placed in Group A alongside Italy, Switzerland and Turkey.
The last time the two sides met in Italy, the Azzurri won 4-0.
Wales won their last meeting with Switzerland, a Euro qualifier in October 2011. Aaron Ramsey and Gareth Bale were the scorers.
The last time Wales played Turkey, in a qualifier for the 1998 World Cup, it ended 6-4 to the Turks with Hakan Sukur scoring four goals.
The draw will see Wales first play Switzerland in Baku, Azerbaijan on the 13th June followed by a match against Turkey, again in Baku, on the 17th June.
Ryan Gigg's side will then travel to Rome for their encounter with Italy on the 21st June 2020.
Wales will play each of their opponents once, with the top two sides in the group progressing into the knockout phase of the competition.
The four best third-placed sides from the six groups will also move into the last-16.
The draw for next year's competition took place in Bucharest, Romania on Saturday afternoon.