  (Abriged) The existence of large-scale and long-lived 2D vortices in
accretion discs has been debated for more than a decade. They appear
spontaneously in several 2D disc simulations and they are known to accelerate
planetesimal formation through a dust trapping process. However, the issue of
the stability of these structures to the imposition of 3D disturbances is still
not fully understood, and it casts doubts on their long term survival.
  Aim: We present new results on the 3D stability of elliptical vortices
embedded in accretion discs, based on a linear analysis and several non-linear
simulations.
  Methods: We derive the linearised equations governing the 3D perturbations in
the core of an elliptical vortex, and we show that they can be reduced to a
Floquet problem. We solve this problem numerically in the astrophysical regime
and we present several analytical limits for which the mechanism responsible
for the instability can be explained. Finally, we compare the results of the
linear analysis to some high resolution simulations.
  Results: We show that most anticyclonic vortices are unstable due to a
resonance between the turnover time and the local epicyclic oscillation period.
In addition, we demonstrate that a strong vertical stratification does not
create any additional stable domain of aspect ratio, but it significantly
reduces growth rates for relatively weak (and therefore elongated) vortices.
  Conclusions: Elliptical vortices are always unstable, whatever the horizontal
or vertical aspect-ratio is. The instability can however be weak and is often
found at small scales, making it difficult to detect in low-order
finite-difference simulations.
