Observational cosmology is in a golden age of discovery, but a deeper understanding of what is meant by a science of cosmology, in the fuller reaches of these words, is in its infancy. It must involve physics, philosophy, and cosmogony - a philosophy of cosmology - and tackle genuinely fundamental questions in cosmology. We will focus on five areas of this kind which are timely, and where the applicants have particular research interests and strengths. Our work will involve both stimulating the pursuit of these questions in the longer term and by a larger audience, and researching fundamental issues in cosmology. These areas are: 1. Issues of measure, and beyond; 2. The structure of space-time; 3. The cosmological constant problem; 4. Entropy, time and complexit; and 5. Symmetries and invariants. Activities will include: mini-courses and master classes by leading cosmologists; a senior visitors' programme; workshops including our US partners; a high-profile international conference; an outreach programme featuring in-depth videoed interviews with key contributors to the field; and a vigorous writing and research programme. Outputs will include: two monographs; research papers; a major international conference; a published collection of original articles; a resource website devoted to Philosophy of Cosmology; and a series of interviews with major figures in cosmology. The enduring impact of the project will be to solve some of the outstanding conceptual problems in the foundations of cosmology, to seed and stimulate future research in the subject and to define it as a major field in philosophy of physics in its own right, with its own distinctive problems and motivations. We understand that progress to this end will best be made by an open-ended enquiry into several philosophical and conceptual questions unique to cosmology, and preferably novel ones, rather than to focus on any single one.