In September 2000, at the Centre de Recerca Matematica in Barcelona, we pre sented a 30-hour Advanced Course on Algebraic Quantum Groups. After the course, we expanded and smoothed out the material presented in the lectures and inte grated it with the background material that we had prepared for the participants; this volume is the result. As our title implies, our aim in the course and in this text is to treat selected algebraic aspects of the subject of quantum groups. Sev eral of the words in the previous sentence call for some elaboration. First, we mean to convey several points by the...