Many of the best-known cryptosystems, particularly public key systems, are numbertheory based and work within the setting of abelian groups. With increased computing power the security of these systems is constantly threatened. Consequently, non-commutative algebraic structures have been investigated as possible sources of cryptographic platforms. We discuss some of the group-theoretic problems exploited in the security of these new cryptosystems. We also describe Hurley's method for encryption and decryption using specially chosen units in group rings.