Polynomial optimization is a fundamental mathematical framework with pervasive applications, typically tackled using Lasserre's Moment-Sum-of-Squares (SOS) hierarchy. Despite its theoretical elegance, the widespread adoption of this hierarchy hinges on overcoming three critical challenges: characterizing when the relaxations are exact, establishing explicit complexity bounds, and scaling the computation to high-dimensional problems. This talk explores recent theoretical and algorithmic advances that address these core bottlenecks.