Topological and geometric ideas are now a mainstay of condensed matter physics, underlying much of our understanding of conventional materials in terms of defects and geometric frustration in ordered media, and protected edge states in topological insulators. In this thesis, I will argue that such an approach successfully identifies the relevant physics in metamaterials and living matter as well, even when traditional techniques fail. I begin with the problem of kirigami mechanics, i.e., designing a pattern of holes in a thin elastic sheet to engineer a specific mechanical response. Using an electrostatic analogy, I show that holes act as sources of geometric incompatibility, a feature that can fruitfully guide design principles for kirigami metamaterials. Next I consider nonequilibrium active matter composed of self-driven interacting units that exhibit large scale collective and emergent behaviour, as commonly seen in living systems. By focusing on active liquid crystals in two dimensions, with both polar and nematic orientational order, I show how broken time-reversal symmetry due to the active drive allows polar flocks on a curved surface to support topologically protected sound modes. In an active nematic, activity instead causes topological disclinations to become spontaneously motile, driving defect unbinding to organize novel phases of defect order and chaos. In all three cases, geometric and topological ideas enable the relevant degrees of freedom to be identified, allowing complex phenomena to be treated in a tractable fashion, with novel and surprising consequences along the way
