Foliation theory grew out of the theory of dynamical systems on manifolds and Ch. Ehresmann's connection theory on fibre bundles. Pion er work was done between 1880 and 1940 by H. Poincare, I. Bendixson, H. Kneser, H. Whitney, and W. Kaplan - to name a few - who all studied "regular curve families" on surfaces, and later by Ch. Ehresmann, G. Reeb, A. Haefliger and ot"ners between 1940 and 1960. Since then the subject has developed from a collection of a few papers to a wide field of research. i owadays, one usually distinguishes between two main branches of foliation theory, the so-called quantitative theory (including homotopy theory and characteristic classes) on the one hand, and the qualitative or geometric theory on the other. The present volume is the first part of a monograph on geometric aspects of foliations. Our intention here is to present some fundamental concepts and results as well as a great number of ideas and examples of various types. The selection of material from only one branch of the theory is conditioned not only by the authors' personal interest but also by the wish to give a systematic and detailed treatment, including complete proofs of all main results. We hope that this goal has been achieved.