polynomial optimization is a class of optimization programs whose objective and constraining functions are polynomials. the core task in polynomial optimization is to compute global optimizers when the optimization is nonconvex. tensor computation is about optimization and decompositions of tensors, such as tensor norms, tensor eigenvalues and tensor decompositions. all these problems are connected to each other by the theory nonnegative polynomials and moment problems. this talk will give an introduction about classical backgrounds, currently existing results and remaining challenges for the research of these topics.