The field of global optimization has advanced significantly over the past three decades. Yet, the solution of even small instances of many nonconvex optimization problems involving the Euclidean norm to global optimality remains beyond the reach of modern global optimization methods. These problems include numerous well-known and high-impact open research questions from a diverse collection of both fundamental and applied fields of science and engineering. In this review, we survey applications in which these problems arise, describe the sources of computational intractability, summarize existing solution methods, and identify promising research directions for future work. We also introduce EuclidLib, a library of instances of optimization problems of this type, to aid in performance benchmarking and algorithm development. The solution of all problems in EuclidLib to global optimality would represent a significant achievement for the field of global optimization and an important contribution to mathematics, science, and engineering.