Crystal Palace have completed the signing of Lille left-back Pape Souare.
Midfielder Stephen McGinn has joined Dundee on loan from Sheffield United until the end of the season.
Milik has penned a five-and-a-half year deal with the Bundesliga club and will join up with his new team-mates at their winter training camp in Portugal on January 3.
"Arkadiusz Milik is considered to be possibly the biggest rising star in Poland at the moment," Leverkusen director Rudi Voller told the club website.
"He has already made it to the Poland squad at the age of 18."
Managing director Wolfgang Holzhauser said: "We have signed Arkadiusz Milik in the face of stiff domestic and international competition."