Uncertainty quantification and its propagation across multi-scale model/experiment chains are key elements of decision-based materials design in the framework of Integrated Computational Materials Engineering. In this context, understanding the sources of uncertainty and their quantification can provide a confidence for the applicability of models for decision making in materials design, which is generally overlooked in the field of materials science. Based on the above-mentioned motivation, different case studies are considered in this work to indicate how Bayesian inverse uncertainty quantification and forward uncertainty propagation approaches operate in various applications and procedure conditions. In this dissertation, inverse uncertainty quantification of model parameters is performed through a Markov Chain Monte Carlo approach; and all propagation of uncertainties from model parameters to model responses are accomplished through the first order second moment approach and/or the forward model analysis of parameters sampled from the posterior probability distribution after the parameter probabilistic calibrations. Moreover, different information fusion approaches are proposed here to smartly combine the probabilistic information obtained from different sources of information for more precise probabilistic predictions of physical systems behaviors. This dissertation starts with the importance of uncertainty quantification for product design and engineering. This is followed by some fundamentals about different sources of uncertainties, different statistical views and approaches for uncertainty quantification and propagation in computational modeling, and the previous work in literature for uncertainty quantification and propagation in materials science problems. Then, uncertainties are evaluated in the case of plastic flow behavior modeling of transformation induced plasticity steels using two different procedures of data training, including sequential training with each experimental data as independent evidence and simultaneous training of all data together as overall evidence. A multi-objective probabilistic calibration of an Ni-Ti precipitation model in MatCalc(C) are also performed against all experimental data simultaneously to quantify the uncertainties of resulting micro-structural features from the model. It should be noted that an empirical relationship for matrix/precipitate inter-facial energy in terms of aging temperature and nominal composition have also been introduced according to the values of inter-facial energy obtained from the model calibration with each given experimental data individually. However, large discrepancies and uncertainties obtained for model results are important reasons to apply co-kriging surrogate modeling for more precise prediction of precipitation behavior and its uncertainty based on the establishment of a linear correlation between the experimental responses and the fitted surrogate model over the results of the precipitation model. In addition, a constrained probabilistic calibration is carried out for a thermo-mechanical model using a distance-based comparison metric of transformation strain-temperature curves. In this case, a design of experiment followed by a variance-based sensitivity analysis are also performed to identify the most influential model parameters before the calibration. Uncertainty quantification in the calculation of Hf-Si binary phase diagram are also discussed in this dissertation. In this case, Bayesian model averaging and an error correlation-based model fusion are also applied to combine all the results obtained from randomly generated models together to make the calculation of phase diagrams more objective rather than being subjective to the expert opinions for the model selection. In the end, the major efforts for uncertainty quantification of thermodynamic properties and phase diagrams are reviewed. For future work, the probabilistic calibration
