In this talk we investigate the Erd\H{o}s-Renyi model for random graphs, $G(n,p)$. Our focus will be on determining the probability that $G(n,p)$ is connected, which was the first problem that Erd\H{o}s and Renyi considered in their original paper. As time permits we will also discuss the ``phase transition'' of $G(n,p)$.