This book focuses on the study of different aspects of the AdS/CFT correspondence, the celebrated conjecture in string theory that relates gauge and gravity theories. It begins with a detailed review of the correspondence, with the main body of work being split into two chapters. The first chapter concerns the evaluation of correlation functions of BPS operators in the AdS/CFT correspondence, as these are useful to make indirect tests of the conjecture and to uncover dynamical features pertaining to the theories in the correspondence. In particular we look at four-point functions in AdS supergravity and their gauge theory interpretation. The second chapter has to do with the application of duality ideas in black hole physics, with emphasis on small black holes. We focus on the fundamental string small black hole and give arguments supporting its study via its CFT dual. The last chapter of the book summarises the content and suggests some interesting directions that could be pursued in the future.