This thesis presents the measurement study of Online Social Networks focusing on Twitter. Twitter is one of the largest social networks using exclusively directed links among accounts. This makes the Twitter social graph much closer to the social graph supporting real life communications than, for instance, Facebook. Therefore, understanding the structure of the Twitter social graph and the way information propagates through it is interesting not only for computer scientists, but also for researchers in other fields, such as sociologists. However, littles is known about the information propagation in Twitter. In the first part, we present an in-depth study of the macroscopic structure of the Twitter social graph. In the second part, we study the propagation of the news media articles shared on Twitter. In the third part we present an experimental study of graph sampling.