Periods are the integrals of algebraic differential forms over domains defined by polynomial inequalities, and are ubiquitous in mathematics and physics. One of the simplest classes of periods are given by multiple zeta values, which are the periods of moduli spaces M_{0,n} of curves of genus zero. They have recently undergone a huge revival of interest, and occur in number theory, the theory of mixed Tate motives, knot invariants, quantum groups, deformation quantization and many more branches of mathematics and physics.Remarkably, it has been observed experimentally that Feynman amplitudes in quantum field theories typically evaluate numerically to multiple zeta values and polylogarithms (which are the iterated integrals on M_{0,n}), and a huge amount of effort is presently devoted to computations of such amplitudes in order to provide predictions for particle collider experiments. A deeper understanding of the reason for the appearance of the same mathematical objects in algebraic geometry and physics is essential to streamline these computations, and ultimately tackle the outstanding problems in particle physics.The proposal has two parts: firstly to undertake a systematic study of the periods and iterated integrals on higher genus moduli spaces M_{g,n} and related varieties, and secondly to relate these fundamental mathematical objects to quantum field theories, bringing to bear modern techniques from algebraic geometry, Hodge theory, and motives to this emerging interdisciplinary area. Part of this would involve the implementation (with the assistance of future postdoc. team members) of an algorithm for the evaluation of Feynman diagrams which is due to the author and goes several orders beyond what has previously been possible, in order eventually to deduce concrete predictions for the Large Hadron Collider.