Understanding the structural intricacies of complex systems, such as hierarchical, fractal, and multiphase materials, presents significant challenges to traditional methods of material characterization. This presentation explores the use of Small- Angle Scattering (SAS), augmented by advanced methodologies including fractal geometry, chaos game representation (CGR), and machine learning (ML), to enhance the analysis of such systems. The fundamental challenges in applying SAS to complex materials are examined, highlighting how novel multi- component scattering models and crossover analysis extend the applicability of SAS to multiphase and hierarchical systems. Application of fractal geometry as a theoretical framework to extract scaling properties from mass and surface fractals is reviewed, emphasizing the differentiation between these fractal types through scattering data. Furthermore, practical applications in biological systems are discussed, specifically the structural analysis of DNA sequences and biological macromolecules, incorporating CGR and multifractal analysis. The integration of ML techniques with SAS is also covered, demonstrating the potential to overcome traditional modeling limitations and optimize structural analysis. Finally, the broader implications of these advances are adressed, including the design of functional materials, enhanced genomic analysis, and applications in radiation materials science.

(Nominated for the JINR Prize for 2024.)