A positive integer n is defined to be cyclic if and only if every group of size n is cyclic. Equivalently, n is cyclic if and only if n is relatively prime to the number of positive integers less than n that are relatively prime to n. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose analogous conjectures about cyclic numbers. Using the 28,488,167 cyclic numbers less than 10⁸, I test the conjectures about cyclic numbers and disprove the cyclic analog of the second conjecture about primes of Hardy and Littlewood. Proofs or disproofs of the remaining conjectures are invited.