This talk is the first in a two-part series on sampling from unnormalized densities and free energy estimation. This part focuses on variance reduction in annealed importance sampling (AIS) through the use of approximate SDE transports. We begin by revisiting AIS and its theoretical connections to the Jarzynski equality and related fluctuation theorems. Building on this foundation, we introduce a general importance sampling identity that enables unbiased estimation in AIS when combined with approximate transports. We characterize the class of optimal transports that minimize estimator variance and show how this framework both subsumes and formalizes the escorted Jarzynski method. In doing so, we provide one of the first formal proofs of its correctness and offer a generalization of Crooks’ fluctuation theorem. This perspective points to new design directions in sampling and inference by also connecting to time reversal and generative modelling, opening the door to many future avenues for the design of neural samplers.