Current Trends in Mathematical Cosmology

We present an elementary account of mathematical cosmology through a series of important unsolved problems. We introduce the fundamental notion of `a cosmology' and focus on the issue of singularities as a theme unifying many current, seemingly unrelated trends of this subject. We discuss problems associated with the definition and asymptotic structure of the notion of cosmological solution and also problems related to the qualification of approximations and to the ranges of validity of given cosmologies.