In this talk we will explore the relatively young field of Extremal Graph Theory. One of the first results in this field answers the following question: how many edges can a graph with n vertices have if it has no complete graph of size k as a subgraph? Since the solution of this problem in the 1940's there have been many generalizations; however overall little is known. One of the main reasons for the lack of knowledge stems from the fact that graphs with certain properties are often hard to construct. Examples of certain known constructions will be given, and possibilities for undergraduate research projects will be discussed.