The Cosmic Web is the fundamental spatial organization of matter on scales of a few up to a hundred Megaparsec, scales at which the Universe still resides in a state of moderate dynamical evolution. galaxies, intergalactic gas and dark matter exist in a wispy weblike spatial arrangement consisting of dense compact clusters, elongated filaments, and sheetlike walls, amidst large near-empty void regions. While the complex intricate structure of the cosmic web contains a wealth of cosmological information, its quantification has remained a major challenge. In this lecture, we describe recent work towards invoking concepts from computational topology and computational geometry to our understanding of the structure of the Cosmic Web and to new insights into the conditions in the primordial Universe out of which it emerged. An important aspect of our understanding of the Cosmic Web concerns the connectivity of the various components. This leads us to recent work on the topological analysis of the Megaparsec scale distribution. To this end, we resort to the homology of the weblike structure, and determine the scale-dependent Betti numbers. To infer this from the discrete spatial galaxy distribution (or of particles in computer models of cosmic structure formation) we extract the Betti numbers from alpha shapes. We have studied the alpha complex of the cosmic weblike point patterns, in order to assess the signature of filaments, walls and voids. Of considerable importance within the context of the multiscale nature of the cosmic mass distrbution, is the information contained in persistence diagrams. Amongst others, I will describe recent work on persistence properties of Gaussian and non-Gaussian random density fields, which form the initial conditions out of which the cosmic web emerged through the force of gravity.