In this paper we show that every L¹-integrable function on ∂Ω can be obtained as the trace of a function of bounded variation in Ω whenever Ω is a domain with regular boundary ∂Ω in a doubling metric measure space. In particular, when Ω supports a 1-Poincaré inequality, the trace class of BV(Ω) is L¹(∂Ω). We also construct a bounded linear extension from a Besov class of functions on ∂Ω to BV(Ω).