Everton will face Portuguese giants Benfica in the Europa League group stage.
The Lisbon club have a strong historical European pedigree having won the European Cup in 1961 and 1962 and reached the final of the UEFA Cup in 1983.
They are no longer top dogs in Portuguese football and finished third last season behind Porto and city rivals Sporting - but should provide a tough test for David Moyes' men.
The Toffees, who beat Czech side Sigma Olomouc 5-1 to reach the group phase, will also face Greek side AEK Athens and BATE Borisov from Belarus in Group I.
Fulham face a trip to Italy to take on Roma and must also tackle Swiss side Basle and CSKA Sofia of Bulgaria in Group E.
The Cottagers overcame Russian side Amkar Perm to qualify for the group phase and now have a trip to the Stadio Olimpico to look forward to to take on Francesco Totti et al.
Celtic, who dropped out of the Champions League at the play-off stage after a 5-1 aggregate defeat to Arsenal, are up against Hamburg, Israeli side Hapoel Tel-Aviv and Rapid Vienna in Group C.
Hamburg finished fifth in the Bundesliga last season under former Tottenham boss Martin Jol, who has now returned to his homeland to coach Ajax. They are second after three games of the new season with two wins and a draw.
Rapid Vienna underlined their strength by beating Aston Villa on away goals in a play-off to reach the group phase.
The first round of matches will be played on Thursday, September 17, with the top two teams from each group progressing to the last 32, where they will be joined by the eight third-placed teams from the Champions League group phase.
Copyright (c) PA Sport 2009, All Rights Reserved.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms