Let $G$ be a cyclic group. There's a theorem which states that if $|G|$ is a prime, then every non-identity member of $G$ is a generator.
What about a cyclic group whose order is not prime: Is there such a group whose every non-identity member is a generator?
Are there other necessary/sufficient conditions regarding groups whose every non-identity member is a generator? (Beyond primality of $|G|$.)