Learning systems often face a critical challenge when applied to settings that differ from those under which they were initially trained. In particular, the assumption that both the source/training and the target/deployment domains follow the same causal mechanisms and observed distributions is commonly violated. This implies that the robustness and convergence guarantees usually expected from these methods are no longer attainable. In this paper, we study these violations through causal lens using the formalism of statistical transportability [Pearl and Bareinboim, 2011] (PB, for short). We start by proving sufficient and necessary graphical conditions under which a probability distribution observed in the source domain can be extrapolated to the target one, where strictly less data is available. We develop the first sound and complete procedure for statistical transportability, which formally closes the problem introduced by PB. Further, we tackle the general challenge of identification of stochastic interventions from observational data [Sec.~4.4, Pearl, 2000]. This problem has been solved in the context of atomic interventions using Pearl's do-calculus, which lacks complete treatment in the stochastic case. We prove completeness of stochastic identification by constructing a reduction of any instance of this problem to an instance of statistical transportability, closing the problem.