We study the topology of random Cech complexes, generated by $n$ iid points in a Euclidean space, and a radius $r$. In particular, we are interested in the limiting behavior of the Betti numbers of such complexes, as $n\to\infty$ and $r\to 0$. We consider different cases in which samples are generated by either pure noise, a probability distribution on a close manifold, or a combination of both. The study of the Betti numbers can be done directly, or via a Morse-theoretic approach, by counting critical points of distance functions. The results show that the Betti numbers of a random complex exhibit a different limiting behavior, depending on both the support of the underlying distribution and the rate in which $r \to 0$. In this talk, we will present the known results to date for each of these limiting phenomena. In addition, we will discuss how these results could be applied to the problem of recovering the topology of a hidden manifold from a finite set of samples.