  We propose a simple criterion to know if an abelian variety $A$ defined over
a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of
rational points; this criterion is based on the endomorphism ring
End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny
class is cyclic, i.e., all its varieties are cyclic; this criterion is based on
the characteristic polynomial of the isogeny class. We find some asymptotic
lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among
certain families of them, when $q$ tends to infinity. Some of these bounds
require an additional hypothesis. In the case of surfaces, we prove that this
hypothesis is achieved and, over all $\mathbb{F}_q$-isogeny classes with
endomorphism algebra being a field and where $q$ is an even power of a prime,
we prove that the one with maximal number of rational points is cyclic and
ordinary.
