The dynamics of three coaxial vortex rings of strengths Gamma(1), Gamma(2)
and Gamma(3) in an ideal fluid is investigated. It is proved that if Gamma(
j), Gamma(j) + Gamma(k) and Gamma(1) + Gamma(2) + Gamma(3) are not zero for
all j, k = 1, 2, 3, then KAM and Poincare-Birkhoff theory can be used to p
rove that if the distances among the rings are sufficiently small compared
to the mean radius of the rings, there are many initial configurations of t
he rings that produce quasiperiodic or periodic motions. Moreover, it is sh
own that the motion become chaotic as the inter-ring distances are increase
d relative to the mean radius.