This is the first of two volumes which present, in a systematic manner, the various areas of research in the field of structural complexity. Since the achievement of a formal definition of the concept of "algorithm", the Mathematical Theory of Computation has developed into a broad and rich discipline. The notion of "complexity of an algorithm" yields an important area of research, known as Complexity Theory, that can be approached from several points of view. The present Volume I is written in a style appropriate for undergraduate students who have taken a first course in Formal Language Theory. The first two chapters of this volume present the basic concepts of structural complexity, providing the background necessary for the understanding of complexity theory. Volume II will be addressed to graduate students and researchers. Both volumes are written in a textbook style; they contain about 200 exercises. The readers are led to a point where very little additional work will enable them to start research projects. In order to ease this step, an effort has been made to point out the main references for each of the results presented in the text.