Recent progress has given numerous examples of spaces whose topology (as measures by homology or cohomology) is of relevance to applications in data, systems, networks, and more. Concomitant with this has been an explosion of computational tools for computing such. This talk considers the evolution of these ideas from spaces to algebraic data structures tethered to a space. The appropriate topological data structure is called a sheaf, and the theory of sheaves yields numerous tools, including cohomology theories, cellular approximations, and more. This talk will outline the basics in the context of numerous examples, including computational issues.