Abstract
We continue our overview of mathematical cosmology with a survey of the third and fourth periods of the development of the subject. The first Part includes the first two periods and is published separately. The third period (1980-2000) continues here with brief descriptions of the main ideas of inflation, the multiverse, quantum, Kaluza-Klein, and string cosmologies, wormholes and baby universes, cosmological stability and modified gravity. The last period, which ends today, includes various more advanced topics such as M-theoretic cosmology, braneworlds, the landscape, topological issues, the measure problem, genericity, dynamical singularities and dark energy. We emphasize certain threads that run throughout the whole period of development of theoretical cosmology and underline their importance in the overall structure of the field. We end this outline with an inclusion of the abstracts of all papers contributed to the second part of the Philosophical Transactions of the Royal Society A, theme issue 'The future of mathematical cosmology'. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.