  A cycle of elliptic curves is a list of elliptic curves over finite fields
such that the number of points on one curve is equal to the size of the field
of definition of the next, in a cyclic way. We study cycles of elliptic curves
in which every curve is pairing-friendly. These have recently found notable
applications in pairing-based cryptography, for instance in improving the
scalability of distributed ledger technologies. We construct a new cycle of
length 4 consisting of MNT curves, and characterize all the possibilities for
cycles consisting of MNT curves. We rule out cycles of length 2 for particular
choices of small embedding degrees. We show that long cycles cannot be
constructed from families of curves with the same complex multiplication
discriminant, and that cycles of composite order elliptic curves cannot exist.
We show that there are no cycles consisting of curves from only the Freeman or
Barreto--Naehrig families.
