Discrete mathematics has seen enormous advances in the last few years, with solutions being found to a number of famous and long-standing questions, such as the Erdos distinct distance problem and the existence conjecture for combinatorial designs. Much of this progress owes to an increased understanding of random and pseudorandom objects. An entire framework, known as the probabilistic method, has grown around the application of randomness to combinatorial problems, while pseudorandomness is playing an increasingly important role.In this proposal, we will consider a range of problems, some stemming from the direct study of random and pseudorandom objects and others arising in areas where randomness and pseudorandomness have proved to be of particular importance. We will be particularly concerned with extensions of the regularity method to sparse graphs and improving bounds for a number of classical problems in graph Ramsey theory. These problems are of a fundamental nature and any progress is likely to lead to new techniques with broader scope for application.