One of the main challenges in Bayesian inference is to efficiently sample from high-dimensional "target" distributions such as the posterior, which is known only through its unnormalized density. A possible approach is to couple a tractible "reference" distribution with the target via a transport map, that approximately pushes forward the reference to the target. In this talk we discuss regularity and approximability of triangular transports for targets on bounded domains. Moreover, we present a multilevel optimization strategy to learn these maps by minimizing the KL-divergence. Here the term "level" may refer for instance to the accuracy of the numerical approximation of the forward map, and solving the corresponding optimization problem in a multilevel fashion can help to decrease computational costs. Finally, we outline how this multilevel strategy is applicable to optimization based inference algorithms more broadly.