The project described in this proposal studies formal proofs and their interaction with computation. The study of propositional proofs is connected to a spectrum of problems in our field, starting with the meta-mathematical quest to explain our failure to understand computation and make progress on the basic questions haunting our field (such as P vs. NP), and ending with the industry-driven quest for better algorithms for solving instances of the satisfiability problem. In a seemingly different direction, the recent introduction of magical probabilistically checkable proofs (PCPs) has opened new horizons in computer science, ranging from a deeper understanding of approximation algorithms and their limits to the construction of super-efficient protocols for the verification of proofs and computations. We suggest to study proofs and computation with three main objectives. First, to construct better SAT solvers via a better understanding of propositional proof systems. Second, to expand the range of applications of PCPs and transform them from the purely theoretical objects that they currently are to practical and accessible formats for use in all settings where proofs are encountered. Third, to expand our theoretical understanding of the intrinsic limits of proofs, with an eye towards explaining why we are unable to make significant progress on central questions in computational complexity. We believe this project can bridge across different regions of computer science such as SAT solving and proof complexity, theory and practice, propositional proofs and probabilistically checkable ones. And its expected impact will start on the theoretical mathematical level that forms the foundation of computer science and percolate to more practical areas of our field.