We will bring you details of who we play when as soon as we know.
Matches will be on 19 September, 3 and 24 October, 7 and 28 November and 12 December.
Anzhi finished third in the Russia Premier League last season and were runners-up in the Russia Cup, losing on penalties in the final to CSKA Moscow. They are currently second-from-bottom of the RPL with two points from six matches.
FC Sheriff are champions of Moldova, taking the title by 14 points last season, winning 25 of 33 matches.
They beat FK Sutjeska of Montenegro 6-1 on aggregate in the second qualifying round of the Champions League before losing to Dinamo Zagreb. That meant a place in play-offs of the Europa League, where they beat Vojvodina of Serbia 3-2 on aggregate. They currently sit second in the Moldova First Division.
Tromso finished fourth in the Norway Premier Division in 2012 to qualify for the first qualifying round of the Europa League.
They beat NK Celje of Slovenia 3-2 on aggregate in the first qualifying round, Inter Baki PIK of Azerbaijan 2-1 on aggregate in the second qualifying round and then Differdange of Luxembourg on penalties after drawing 1-1 on aggregate in the third qualifying round.
That took them into the play-offs against Besiktas of Turkey. After being eliminated 3-2 on aggregate over the two legs, Tromso were re-instated into the competition after UEFA excluded Besiktas for a year.
Tromso are currently 13th of 16 in Norway after 21 matches of the domestic campaign.