I will present a new topological formalism that, for the first time, describes topology as a multi-scale concept. This has a direct and strong relevance to the topological analysis of structure formation process in the cosmos, given that this proceeds in a hierarchical fashion. Rooted in algebraic topology, the concepts I will describe stem from (persistence) homology and Morse theory. Although the mathematical theory behind the concepts have been known for over a century, only recently have they become of practical relevance, due to breakthroughs in computational topology over the last decade. Taking gaussian random fields and the web-like spatial distribution of matter in the Universe as running examples, I will demonstrate that the formalism allows us to describe topology at a level of detail that far supersedes those provided by the standard topological descriptors like Euler characteristic and genus. In this context, I will introduce persistence and ratio landscapes as an empirical statistical description of persistence homology. They prove to be powerful tools to discriminate between the spatial structure emanating in different cosmologies. Most promising is our recent realization that it provides for a novel way of hunting for primordial non-gaussianities. In a related aspect, I will present a software for interactive visualization of the hierarchical topological structures. It exploits the geometric aspects of Morse theory, to detect, describe and quantify the filamentary patterns of the Cosmic web. The software is a promising tool applicable to the analysis of structural patterns in a wide range of astronomical areas of interest -- the detection and characterization of stellar streams at galactic scales being a promising candidate.