We discuss a new construction of a solution to the measure transport problem. This solution is defined as a zero of an infinite-dimensional score-matching problem. We develop an infinite-dimensional generalization of a Newton method to find this zero, which also serves as its constructive existence proof. We define a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The Newton method is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map. It is appropriate for the variational inference setting, where the score is known, and for sampling certain chaotic dynamical systems, where a conditional score can be calculated even in the absence of a statistical model for the target. Fast computation of scores in this setting is discussed along with the roadmap to applying the transport algorithm to Bayesian filtering. We take a brief detour to discuss the application of score computation for the linear response problem in chaotic systems, which is joint work with Qiqi Wang. The transport construction is joint work with Youssef Marzouk.