Improved measurement capabilities and computational resources have led to significant algorithmic developments toward efficient uncertainty quantification (UQ) for computational models. Models of physical systems often involve input parameters that exhibit certain degree of uncertainty. Estimation and propagation of these uncertainties are crucial for model validation, computational/experimental design and decision making. This talk will focus on probabilistic methods with emphasis on Polynomial Chaos (PC) expansions as a means for functional representation of random variables. The talk will highlight the use of PC methods both for forward propagation of uncertainties and for inverse problems, such as parameter estimation via Bayesian inference. I will list associated major challenges, including the curse of dimensionality and model structural error estimation, in the context of computationally expensive models of physical systems. Both fundamental and more recent methods will be introduced and demonstrated, impacting a range of applications.