Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr [Albert Einstein: Philosopher-Scientist, Library of Living Philosophers Vol. VII, edited by P. A. Schilpp (Open Court, 1998), pp. 199-241] and later realized more technically by Kochen and Specker [J. Math. Mech. 17, 59 (1967)]. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves in Isham and Butterfield [Int. J. Theor. Phys. 37, 2669 (1998)]. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, Wigner's theorem, Gleason's theorem, and Bell's theorem also relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.