Designing novel materials with superior and resilient mechanical properties requires precise tailoring of their structures. In particular, the design of architected metamaterials, which feature carefully engineered geometries, represents a promising path given their versatility. Their structural arrangements are typically highly complex, characterized by nontrivial and often disordered configurations. Despite this complexity, their geometries must be represented in a numerically tractable manner to enable practical applications in data-driven materials science. Consequently, identifying suitable morphological descriptors for data analysis and algorithmic modeling presents a significant challenge. In this context, novel methods from applied topology emerge as powerful tools to address this challenge. Among them, persistent homology enables extracting and representing local geometrical features as point clouds, effectively converting complex structural information into simple data. This transformation facilitates the application of data science techniques, including statistical analysis and machine learning models, to gain deeper insights into material structures. Here, we discuss the application of persistent homology to characterize correlated disordered structures. Focusing first on the prototypical case of point patterns, we show that persistent homology efficiently describes structural properties that only appear on large scales, such as hyperuniformity. We further illustrate that this description is exploitable for inferring nontrivial structural characteristics of patterns in a data-driven fashion. Finally, we explore promising extensions of this framework to field-based descriptions, broadening its applicability to the mechanical engineering of metamaterials. This includes structures such as spinodoid architectures, where topological analysis can provide valuable insights into their design and performance.