This book is about relations between three di?erent areas of mathematics and theoreticalcomputer science: combinatorialgroup theory cryptography and c- plexity theory. We explorehownon-commutative(in?nite) groups which arety- callystudiedincombinatorialgrouptheory canbeusedinpublickeycryptography. We also show that there is a remarkable feedback from cryptography to com- natorial group theory because some of the problems motivated by cryptography appear to be new to group theory and they open many interesting research - enues within group theory. Then we employ complexity theory notably generic case complexity of algorithms for cryptanalysisof various cryptographicprotocols based on in?nite groups. We also use the ideas and machinery from the theory of generic case complexity to study asymptotically dominant properties of some in?nite groups that have been used in public key cryptography so far. It turns out that for a relevant cryptographic scheme to be secure it is essential that keys are selected from a very small (relative to the whole group say) subset rather than from the whole group. Detecting these subsets ( black holes ) for a part- ular cryptographic scheme is usually a very challenging problem but it holds the keyto creatingsecurecryptographicprimitives basedonin?nite non-commutative groups. The book isbased onlecture notesfor the Advanced Courseon Group-Based CryptographyheldattheCRM BarcelonainMay2007. Itisagreatpleasureforus to thank Manuel Castellet the HonoraryDirector of the CRM for supporting the idea of this Advanced Course. We are also grateful to the current CRM Director JoaquimBruna and to the friendly CRM sta? especially Mrs. N. PortetandMrs. N. Hern´ andez for their help in running the Advanced Course and in preparing the lecture notes.