We describe in this work the BASS survey for brown dwarfs in young moving groups of the solar neighborhood, and summarize the results that it generated. These include the discovery of the 2MASS J01033563–5515561 (AB)b and 2MASS J02192210–3925225 B young companions near the deuterium-burning limit as well as 44 new low-mass stars and 69 new brown dwarfs with a spectroscopically confirmed low gravity. Among those, ~20 have estimated masses within the planetary regime, one is a new L4 γ bona fide member of AB Doradus, three are TW Hydrae candidates with later spectral types (L1–L4) than all of its previously known members and six are among the first contenders for low-gravity ≥ L5 β/γ brown dwarfs, reminiscent of WISEP J004701.06+680352.1, PSO J318.5338–22.8603 and VHS J125601.92–125723.9 b. Finally, we describe a future version of this survey, BASS-Ultracool, that will specifically target ≥ L5 candidate members of young moving groups. First experimentations in designing the survey have already led to the discovery of a new T dwarf bona fide member of AB Doradus, as well as the serendipitous discoveries of an L9 subdwarf and an L5 + T5 brown dwarf binary.