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...