Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number
$S$
for the onset of the bubble (
$S^*_B$
) and the cone (
$S^*_C$
) is determined as the jet-to-ambient density ratio
$\varLambda$
is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts
$S^*_B$
that decreases with increasing values of
$\varLambda$
for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at
$S^*_C$
, found to be independent of
$\varLambda$
, in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number
$Re_{eff}$
, defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at
$Re_{eff}=200$
for all
$\varLambda$
. Similar to the theoretical predictions, numerical calculations of
$S^*_B$
decrease monotonically as
$\varLambda$
is increased. The critical swirl numbers for the cone,
$S^*_C$
, are found to depend strongly on viscous effects; for
$\varLambda =1/5$
, the low jet Reynolds number (51) at
$Re_{eff}=200$
delays the transition to the cone, while for
$\varLambda =5$
at
$Re_{eff}=200$
, the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing
$S^*_C$
.