The principal purpose of this introductory chapter is to acquaint the reader with the terminology and basic facts of abelian groups which will be used throughout the text. Some of the proofs will be omitted as they are standard and can be found in textbooks on algebra or on group theory. The fundamental types of groups, together with their main properties, are briefly discussed here. We shall save numerous repetitions by the adoption of their conventional notations. Maps, diagrams, categories, and functors are also presented; they will play an important role in our developments. Some of the most useful topologies in abelian groups will also be surveyed. A reader not familiar with the subject treated here is advised to read this chapter most carefully.