Output of an engineering system has a distribution due to its parameters uncertainty. Therefore, probability of failure will not be zero even by applying a high safety factor in a nominal design which cause some cases to fail in some given criterion after fabrication and test in the real world. Therefore, it is necessary to implement an uncertainty analysis and reliability estimation in order to achieve a certain level of reliability and probability of failure especially during the design of consequential systems. While many numerical methods have been developed for uncertainty propagation in probabilistic framework, much less has been discussed in the nonprobabilistic framework, which is important whenever one does not possess sufficient data or knowledge of the underlying systems. The main goal of this thesis is to apply fuzzy algebra to model uncertainty and their propagation through complex physical systems in order to quantify their effects on performance and reliability. The secondary goal of this thesis is to apply the proposed fuzzy approaches in time varying systems in order to assess their remaining life with a certain level of reliability. In particular, this thesis develops fuzzy numerical methods by modification of conventional ?-cut method with the aim of attaining high and reliable accuracy. ?-cut method is very efficient from computational effort points of view, but it has less accuracy in cases with high nonlinear effects and high mutual correlation between its parameters; In addition, estimation accuracy will not improve to a reliable level with the increase in mesh number of discretization. Therefore, implementation of ?-cut method is confined to several applications such as finite element analysis. Modified ?-cut method is proposed which is a fuzzy representation of discretized monte carlo approach in uncertainty quantification. Therefore, with the increase in the mesh number of discretization, estimated output distribution will tend to exact solution. Despite the ?-cut method, modified method discretized the input distribution in a partitioned way and uses different extension principle. Unfortunately, computational cost in the modified ?-cut method will exponentially increase with the increase in the number of uncertain input variables. Modified ?-cut method has a good accuracy and performance in systems which have little number of uncertain parameters as illustrated in several case studies in the thesis. Monte carlo simulation is applied on each problem and its results is considered as exact solution of the problem. Results of Modified ?-cut method is compared to ?-cut method and results obtained from monte carlo simulation as verification which shows that Modified ?-cut method increases the accuracy of estimation with increase in the computational cost reasonably. Then hybrid ?-cut method is proposed for analyzing systems with many uncertain parameters. The hybrid approach is based on two ?-cut method and modified ?-cut method. In this method, input parameters are divided into two groups due to their sensitivity and nonlinearity and discretized parameters combine in two inner and outer loop. The parameters with higher level of sensitivity will combine with modified ?-cut method in inner loop, and the parameters in the other group will combine with ?-cut method in outer loop. Finally results will combine and assemble using operators of modified ?-cut method. Hybrid ?-cut method has a good accuracy and performance in systems which have many uncertain parameters, but little number of uncertain parameters has a high level of sensitivity as illustrated in several case studies in the thesis. Results of Hybrid approach is compared with ?-cut method and monte carlo simulation as verification which shows that hybrid ?-cut method increases the accuracy of estimation with increase in the computational cost reasonably. In the end, application of fuzzy approaches on reliability analysis of systems varying with time is studied with the aim of estimation of remaining life time with a certain probability of failure. For this aim, a fatigue crack growth problem is considered and Estimated life is compared with the life obtained from monte carlo method and the results has a good agreement. Keywords: Uncertainty quantification, Reliability analysis, ?-cut method, Modified ?-cut method, Hybrid ?-cut method, Fuzzy algebra.