  We derive a relationship for the vortex aspect ratio $\alpha$ (vertical
half-thickness over horizontal length scale) for steady and slowly evolving
vortices in rotating stratified fluids, as a function of the Brunt-Vaisala
frequencies within the vortex $N_c$ and in the background fluid outside the
vortex $\bar{N}$, the Coriolis parameter $f$, and the Rossby number $Ro$ of the
vortex: $\alpha^2 = Ro(1+Ro) f^2/(N_c^2-\bar{N}^2)$. This relation is valid for
cyclones and anticyclones in either the cyclostrophic or geostrophic regimes;
it works with vortices in Boussinesq fluids or ideal gases, and the background
density gradient need not be uniform. Our relation for $\alpha$ has many
consequences for equilibrium vortices in rotating stratified flows. For
example, cyclones must have $N_c^2 > \bar{N}^2$; weak anticyclones (with $|Ro|
< 1$) must have $N_c^2 < \bar{N}^2; and strong anticyclones must have $N_c^2 >
\bar{N}^2$. We verify our relation for $\alpha$ with numerical simulations of
the three-dimensional Boussinesq equations for a wide variety of vortices,
including: vortices that are initially in (dissipationless) equilibrium and
then evolve due to an imposed weak viscous dissipation or density radiation;
anticyclones created by the geostrophic adjustment of a patch of locally mixed
density; cyclones created by fluid suction from a small localised region;
vortices created from the remnants of the violent breakups of columnar
vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios
of our numerically-computed vortices validate our relationship for $\alpha$,
and generally they differ significantly from the values obtained from the
much-cited conjecture that $\alpha = f/\bar{N}$ in quasi-geostrophic vortices.
