Singularities in Inflationary Cosmology:. a Review

We review here some recent results that show that inflationary cosmological models must contain initial singularities. We also present a new singularity theorem. The question of the initial singularity re-emerges in inflationary cosmology because inflation is known to be generically futureeternal, It is natural to ask, therefore, if inflationary models can be continued into the infinite past in a non-singular way. The results that we discuss show that the answer to the question is “no.” This means that we cannot use inflation as a way of avoiding the question of the birth of the Universe. We also argue that our new theorem suggests — in a sense that we explain in the paper — that the Universe cannot be infinitely old.