In my talk on random graphs, we will go through what a random graph is and the two variations, the Gilbert graphs, and the Erdos-Renyi model. We will discuss the history of these models and how they are similar, but also different. From there we will discuss the properties of random graphs, and real-world applications of these random graph models to lead us to conclude the talk where we will discuss the small world problem. This problem is the hypothesis that no person is separated by more than six degrees from another individual. This problem can be illustrated by the Watts-Strogatz model which is a random graph that shows network properties.