when do the lebesgue-bochner function spaces contain a copy or a complemented copy of any of the classical sequence spaces? this problem and the analogous one for vector- valued continuous function spaces have attracted quite a lot of research activity in the last twenty-five years. the aim of this monograph is to give a detailed exposition of the answers to these questions, providing a unified and self-contained treatment. it presents a great number of results, methods and techniques, which are useful for any researcher in banach spaces and, in general, in functional analysis. this book is written at a graduate student level, assuming the basics in banach space theory.