Let E be a real, separable Banach space and denote by $L^0(\Omega ,E)$ the space of all E-valued random vectors defined on the probability space Ω. The following result is proved. There exists an extension $\tilde{\Omega }$ of Ω, and a filtration ${\left({\widetilde{ℱ}}_t\right)}_{t\ge 0}$ on $\tilde{\Omega }$, such that for every $X\in L^0(\Omega ,E)$ there is an E-valued, continuous $\left({\widetilde{ℱ}}_t\right)$-martingale ${\left(M_t\left(X\right)\right)}_{t\ge 0}$ in which X is embedded in the sense that $X=M_{\tau }\left(X\right)$ a.s. for an a.s. finite stopping time τ. For E = ℝ this gives a Skorokhod embedding for all $X\in L^0(\Omega ,ℝ)$, and for general E this leads to a representation of random vectors as...