Accurate parameter estimation is a challenging task that demands realistic models and algorithms to obtain the parameter’s probability distributions. The Bayesian theorem in conjunction with sampling methods proved to be invaluable here since it allows for the formulation of the problem in a probabilistic framework. This opens up the possibilities of using prior information and knowledge about parameter distributions as well as the natural incorporation of aleatory and epistemic uncertainties. Traditionally, Markov Chain Monte Carlo (MCMC) methods are used to approximate the posterior distribution of samples given some data. However, these methods usually need a large amount of samples and therefore a large amount of model evaluations. Recent advances in transport theory and its application in the context of Bayesian model updating (BMU) make it possible to approximate the posterior distribution analytically and hence eliminate the need for sampling methods. This paves the way for the usage in real-time applications and for fast parameter estimation. We investigate here the application of transport maps to a simple analytical model as well as a structural dynamics model. The performance is compared to an MCMC approach to assess the accuracy and efficiency of transport maps. A discussion about requirements for the implementation of transport maps as well as details on the implementation are also given.