When B36 was founded, they battled with local rivals for the rights to the Gundadalur Stadium, although both teams used the venue and played a number of derby matches there during that initial season.

To make matters worse, rival team HB Tórshavn experienced a lot of success in the same period.

The results improved, and the cup was won in 1991, although the National Championship was lost in the final match that same year.

Despite being down with 2 men, B36 Tórshavn won the match 3–2, when club legend Róaldur Jacobsen scored from 72 meters in the last minute.

Likewise, the former Faroe Islands national under-21 football team player Meinhard Olsen was sold to Kristiansund BK in 2018.

In June 2019, he made his debut on the Faroe Islands national football team in a match against Spain.

Most recently, Andrass Johansen, Hannes Agnarsson, Jann Benjaminsen and Magnus Egilsson were called up for the matches against Liechtenstein and Denmark.

Andrias Eriksen, Eli Nielsen, Hannes Agnarsson, Benjamin Heinesen, Andrass Johansen, Jann Benjaminsen, Símun Sólheim and Silas Eyðsteinsson have all played on The Faroese U21 National Team.

In 2005/06, the team reached the second round in the UEFA Cup after defeating ÍBV Vestmannaeyar of Iceland 3–2 on aggregate.

In the second round, they narrowly lost to Danish Superliga-club FC Midtjylland 1–2 in the away leg and drew 2–2 at home.

The team also made it through the first qualifying round by defeating OFK Titograd from Montenegro after a 0–0 draw at home and a 2–1 victory away.

In the 2023–24 Conference League, B36 made it to the third qualifying round after victories against Paide Linnameeskond from Estonia and Haverfordwest County from Wales.

Note: Flags indicate national team as defined under FIFA eligibility rules; some limited exceptions apply.

Fully up to date as of match played 18 July 2024 In the 2020–21 season B36 became the first Faroese team to get through three rounds of a UEFA club competition.