Approximation Methods for Polynomial Optimization : Models Algorithms and Applications 1. Introduction. 2. Polynomial over the Euclidean Ball 3. Extensions of the Constraint Sets 4. Applications 5. Concluding Remarks. Polynomial optimization have been a hot research topic for the past few years and its applications range from Operations Research biomedical engineering investment science to quantum mechanics linear algebra and signal processing among many others. In this brief the authors discuss some important subclasses of polynomial optimization models arising from various applications with a focus on approximations algorithms with guaranteed worst case performance analysis. The brief presents a clear view of the basic ideas underlying the design of such algorithms and the benefits are highlighted by illustrative examples showing the possible applications. This timely treatise will appeal to researchers and graduate students in the fields of optimization computational mathematics Operations Research industrial engineering and computer science.