The classical models of random graphs were introduced by Erd?s, Rényi and Gilbert in the late 1950s. One of the initial motivations for these random models was to provide a new source of examples or counterexamples in the theory of graphs that had remained elusive to other deterministic approaches. Over the past decades, random graphs have become a subject of active research on their own merit due to their fascinating properties, the beauty of the mathematical methods involved and also the applications and connections to computer science and statistical physics. As a result, the field has seen a dramatic explosion in the number of papers, new models and open questions. In this talk I will focus on the existence of spanning structures (such as perfect matchings or Hamilton cycles) in random graphs, which is one the main themes of my research, and will give a gentle overview on some of the main ideas and tools in this field.