Soft Computing (SC) represents a significant paradigm shift in the aims of computing, which reflects the fact that the human mind, unlike present day computers, possesses a remarkable ability to store and process information which is pervasively imprecise, uncertain and lacking in categoricity. At this juncture, the principal constituents of Soft Computing (SC) are: Fuzzy Systems (FS), including Fuzzy Logic (FL); Evolutionary Computation (EC), including Genetic Algorithms (GA); Neural Networks (NN), including Neural Computing (NC); Machine Learning (ML); and Probabilistic Reasoning (PR). In this work, we focus on fuzzy methodologies and fuzzy systems, as they bring basic ideas to other SC methodologies. The other constituents of SC are also briefly surveyed here but for details we refer to the existing vast literature. In Part 1 we present an overview of developments in the individual parts of SC. For each constituent of SC we overview its background, main problems, methodologies and recent developments. We focus mainly on Fuzzy Systems, for which the main literature, main professional journals and other relevant information is also supplied. The other constituencies of SC are reviewed shortly. In Part 2 we investigate some fuzzy optimization systems. First, we investigate Fuzzy Sets-we define fuzzy sets within the classical set theory by nested families of sets, and discuss how this concept is related to the usual definition by membership functions. Further, we will bring some important applications of the theory based on generalizations of concave functions. We study a decision problem, ie the problem to find a” best” decision in the set of feasible alternatives with respect to several (ie more than one) criteria functions. Within the framework of such a decision situation, we deal with the existence and mutual relationships of three kinds of” optimal decisions”: Weak Pareto-Maximizers, Pareto-Maximizers and Strong Pareto-Maximizers-particular alternatives satisfying some natural and rational conditions. We also study the compromise decisions maximizing some aggregation of the criteria. The criteria considered here will be functions defined on the set of feasible alternatives with the values in the unit interval. In Fuzzy mathematical programming problems (FMP) the values of the objective function describe effects from choices of the alternatives. Among others we show that the class of all MP problems with (crisp) parameters can be naturally embedded into the class of FMP problems with fuzzy parameters. Finally, we deal with a class of fuzzy linear programming problems. We show that the class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems we define the concept of duality and prove the weak and strong duality theorems. Further, we investigate special classes of FLP-interval LP problems, flexible LP problems, LP problems with interactive coefficients and LP problems with centered coefficients. We present here an original mathematically oriented and unified approach.