  An elementary survey of mathematical cosmology is presented. We cover certain
key ideas and developments in a qualitative way, from the time of the Einstein
static universe in 1917 until today. We divide our presentation into four main
parts, the first part containing important cosmologies discovered until 1960.
The second period (1960-80) contains discussions of geometric extensions of the
standard cosmology, singularities, chaotic behaviour, and the initial input of
particle physics ideas into cosmology. Our survey for the third period
(1980-2000) continues 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 Philosophical Transactions of the Royal Society A, Theme
Issue `The Future of Mathematical Cosmology'.

History and Philosophy of Physics;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory