Let E be a real, separable Banach space and denote by L0(Ω,E) the space of all E-valued random vectors defined on the probability space Ω. The following result is proved. There exists an extension Ω of Ω, and a filtration (Ft)t≥0 on Ω, such that for every X∈L0(Ω,E) there is an E-valued, continuous (Ft)-martingale (Mt(X))t≥0 in which X is embedded in the sense that X=Mτ(X) a.s. for an a.s. finite stopping time τ. For E = ℝ this gives a Skorokhod embedding for all X∈L0(Ω,R), and for general E this leads to a representation of random vectors as stochastic integrals relative to a Brownian motion