In this talk, we will cover the basics of measurable cardinals and their relationship to non-trivial elementary embeddings.  We proceed with basic facts about the constructible universe, L.  After laying this groundwork, we show L cannot have a measurable cardinal.  Time permitting, we will discuss the dichotomy introduced by Jensen's covering lemma: either L is a good approximation to V, or there is a non-trivial elementary embedding from L to L.