We shall give an introduction to subnormal operators, beginningwith normal operators and the spectral theorem. We will discuss a new C*characterization of subnormality. We also investigate subnormal operatorsthat are essentially normal; including some conditions for essentialnormality. Also some new results will be given, obtained from C*-algebratechniques, about self commutators of operators that are not essentiallynormal. Lastly, we will discuss when two operators generate isomorphicC*-algebras.