In this talk I will survey some of the cryptosystems based on group theoretic problems and their computational complexity such as Conjugacy, Membership, Endomorphism, Word, Twisted Conjugacy, and Geodesic Length Problems. I will speak about some non-abelian groups that have been proposed as platforms for such cryptosystems: Braid, Polycyclic, Metabelian, Grigorchuk, Thompson, Matrix, Hyperbolic, Small Cancellation, right angled Artin Groups and free nilpotent p-groups. The focus of the talk will be on infinite polycyclic group-based cryptosystems as well as a cryptosystem based on semidirect product of (semi)-groups. The latter is a joint work with V. Shpilrain. There will be open problems related to both computational complexity of group theoretic problems and cryptographic problems that I will mention at the end of the talk. The talk will be accessible to graduate students in computer science and mathematics.