Barcelona will again face seven-time winner AC Milan in the Champions League, and Bayern Munich is in a group which includes Manchester City.
Arsenal face a daunting task after being drawn alongside last season's runners-up as well as Rafael Benitez's Napoli and Marseille.
The Gunners, who qualified after completing a comprehensive play-off win over Fenerbahce this week, were placed in the tough-looking Group F at UEFA's draw in Monaco.
Celtic, who famously beat Barcelona last season, will face the Catalan giants again along with two other former winners in Group H in AC Milan and Ajax.
Manchester City will again play reigning champions Bayern Munich - who they faced two years ago - CSKA Moscow and Viktoria Plzen in Group D.
Chelsea face two of the sides they conquered en route to last year's Europa League success in Basle and Steaua Bucharest, as well as Schalke, in Group E.
David Moyes' Champions League bow with Manchester United will come against Shakhtar Donetsk, Bayer Leverkusen and Real Sociedad in Group A.
Group A: Manchester United, Shakhtar Donetsk, Bayer Leverkusen, Real Sociedad
Group B: Real Madrid, Juventus, Galatasaray, FC Copenhagen
Group C: Benfica, Paris St Germain, Olympiakos, Anderlecht
Group D: Bayern Munich, CSKA Moscow, Manchester City, Viktoria Plzen
Group E: Chelsea, Schalke, Basel, Steaua Bucharest
Group F: Arsenal, Marseille, Borussia Dortmund, Napoli
Group G: Porto, Atletico Madrid, Zenit St Petersburg, Austria Vienna
Group H: Barcelona, AC Milan, Ajax, Celtic
FOXSoccer.com's newswire services contributed to this report.
