In this dissertation class groups and unit groups of number fields with elementary Galois groups of order 4 and 8 are considered. In chapter 3 we consider bicyclic biquadratic extensions K/k and give a method for determining the structure of the 2-class group of K. In chapters 4 and 5 this method is applied to real and imaginary bicyclic biquadratic extensions of Q. In chapter 6 a method for determining the unit group of an imaginary octic field is given. In the final chapter all imaginary octic fields of class number less than or equal to 16 or prime class number are determined.