The past few years have witnessed exciting discoveries in different areas of coding theory and computational number theory. The present monograph, which exhibits the author's "Habilitationsschrift," is a collection of five different topics dealing with these two important fields. We will start with a purely coding theoretic question and finish with a discussion of some problems from computational number theory. Along the way, we will gradually change our focus from coding theory to number theory. Our emphasis is almost entirely on the development of fast and practical algorithms for the problems involved. In many cases it turns out that having a view for both number theory and coding theory is a clear advantage. This is best demonstrated by Chapters 2 and 3, where we encounter most of the interrelations between coding and number theory